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LECTURES  ON   MATHEMATICS 


THK    BOSTON    COLLOQUIUM 


Lectures  on  Mathematics 


DELIVERED  FROM  SEPTEMBER  2  TO  •',,  I9()f5,  BEFORE 
MEMBERS  OF  THE  AMERICAN  MATHEMATICAL  SOCIEIY 
IN  CONNECTION  WITH  THE  SUMMER  MEETINC  HELD 
AT  THE  MASSACHUSETTS  INSTITUTE  OF  TECHN0L0(;Y 
I50ST(JN,     MASS. 


EDWARD    BURR    VAN    VLKCK 

HENRY    SEELY    WHITE 

FREDERICK    SHICNSTONE    WOODS 


XfU)  York 

PTTILISHFI)    FOR 

THE    AMERICAN    >L\  ITnALVlTCAL    SOCIETY 
r.v 

THE    MACMIEEAN    COMPANY 

LONDON:     MACMII.T.AN    &    CO.,    Ltd. 
1905 


CorvRHiiiT,  190') 
Bv  THE    MACMILLAN    COMPANY 


L»>c»5t[«.  Pa. 


10 

PROFK.^bOK    JOHN    MOXROE    VAN    VLECK,    LL.D. 

rur.SK    LECTURES    ARE    AFFECTIOXATELY    INSCRIBED    BY 
HIS    FORMER    PUPILS 

HENRY    SEELY    WHITE 

EDWARD    BURR    VAN    VLECK 

FREDERICK    SHENSTONE    WOODS 


PREFACE. 

Foil  a  number  of  years  the  American  Mathematical  Society  lias 
held  a  Colloquimn  in  connection  with  its  Summer  Meeting  at  in- 
tervals of  two  or  three  years.  In  the  circular  sent  out  pri(jr  to  the 
first  Collo(|uiiun,  in  1896,  the  purpose  and  the  plan  of  the  under- 
taking were  described  as  follows:^  "The  objects  now  attained 
by  the  Summer  Meeting  are  two-fold  :  an  opportunity  is  offered 
for  ])resenting  before  discriminating  and  interested  auditors  the 
results  of  research  in  special  fields,  and  personal  acquaintance  and 
mutual  helpfulness  are  promoted  among  the  members  in  attend- 
ance. These  two  are  the  prime  objects  of  such  a  gathering.  It 
is  believed  however  that  a  third  no  less  desirable  result  lies  within 
reach.  From  the  concise,  unrelated  papers  presented  at  any 
meeting  only  few  derive  substantial  benefit.  The  mind  of  the 
hearer  is  too  unprepared,  the  impression  is  of  too  short  duration 
to  produce  accurate  knowledge  of  either  the  content  or  the  method. 
.  .  .  Positive  and  exact  knowledge,  scientific  knowledge,  is  rarely 
increased  in  these  short  and  stimulating  sessions. 

''On  the  other  hand,  the  courses  of  lectures  in  our  best  univer- 
sities, even  with  topics  changing  at  intervals  of  a  few  weeks,  do 
give  exact  knowledge  and  furnish  a  substantial  basis  for  reading 
and  investio-ation.   .   .   . 


Cf.  Bull.  Am.  Math.  Son.,  ser.  2,  vol.  P,  (1890),  p.  49. 


vni  PREFACE. 

"To  extend  the  time  of  a  lecture  to  two  liours,  and  to  multiply 
tliis  time  by  three  or  bv  six,  would  be  practicable  within  the  limits 
of  one  week.  An  expert  lecturer  could  present,  in  six  two-hour 
lectures,  a  moderately  extensive  chapter  in  some  one  branch  of 
mathematics.  With  some  new  matter,  much  that  is  old  could  be 
mingled,  including  for  example  digests  of  recent  or  too  much 
neglected  publications.  There  would  be  time  for  some  elementary 
details  as  well  as  for  more  profound  discussions.  In  short,  lectures 
could  be  made  profitable  to  all  who  have  a  general  knowledge  of 
the  higher  mathematics." 

As  a  forerunner  of  the  Colloquia  here  outlined  may  be  men- 
tioned the  Evanston  Colkxpiium  of  189-'^),  which  followed  the 
Congress  of  Mathematics  held  in  connection  with  the  A\'orld's 
Fair  in  C'liicago,  ]*rofessor  Klein,  of  Gottingen,  being  the  sole 
sj)eaker.  But  whereas  that  Colloquium  covered,  in  a  descriptive 
manner,  a  variety  of  topics,  —  it  comprised  twelve  lectures, — 
tile  Collo(|uia  of  the  Society  have  l)een  characterized  l)y  close  con- 
tact witii  the  actual  analytical  develo})ment    of  the  to})ic  treated. 

The  following  Colhxpiia  have  been  iield  : 

I.       TlIH    Bl'FFALO    C<)I>L(KJUirM,    1 89G. 

(</)    I^rofcssor  Maximk  iVxiiKU,  of  Harvard  University  :   "Lin- 
eal- Diiferential  I*>(puUions,  and  Their  A j)plications." 
This  ('olhxjuium  has  not  been  j)ublisli(Hl,  I)ut  several  papers 
aj)i)eare<l   at  about   the  time  of  the   Colh)(|uium,  in  which   the 
autlior  dealt  with  to|)i('s  treated  in  the  lectures.* 

(i>)  I'roft's-oi-  .Iamks    Pii;iii'()NT,  of  Yale  Cniversity  :     "  Galois's 
Thcoi'v  of  I'^jiiat ions." 

'I'his   CoIl'Kjuium  was    jiublislied    in    the    Ainur/.s  of  }faf/ic- 
iiuiUrs,  ser.  '2,  vols.    1  and  1^   (  1!M)0). 

■  'I'wo  of  these  |i;iprrs  wtTt' ;  "Ite^'ular  Pciints  (if  I>iiU';ir  1  )illerential  lujiKi- 
tiuii-  nt'  the  Sec. ,11'!  <  )nlei-"  :  Harvanl  I 'iiiversity,  Is;)!',;  "  Notes  on  Smiit.'  Points 
in  the  Theory  ot'  Linear  I  )ilVereiitial   Iviuations,"   Amuih  of  Math.,  vol.   \'l,   18'JS. 


i'in-:FAc'K 


IX 


IT.     Tin-:  CA:\n!i;iiK;j-:  Com-oc^i'iim,   18iiS. 

(a)   l^rofessor    AViLiJA^r    F.    Osgood,    of    Harvard    University: 
''Selected  Topics  in  the  Theory  of  Fnnctions." 
This    colhxjninm    was    pnblished    in    tiie    IhiUdin    of    f/ie 
Ainer.  Math.  Soc,  ser.  2,  voh  o  (1S9S),  p.  59. 

(/>)  Professor  AirniUK  G.  Wei'.ster,  oi'  Clark  University  :  "The 
Partial  Differential  Equations  of  A\'ave  l*ropagation." 

Ill,       ThK    PrHACA    CoLT-OQUIU^r,    1901. 

{<i)  Professor  Oscar  Bot.za,  of  the  University  of  Chicago  :  "  The 
Sini])lest  Type  of  I'roblems  in  the  Calculus  of  Variations." 
Published   in  amplified  form   under  the  title  :   Lectures  on 
fJie  Ca/cuhis  of  1  a/'/«//o/;N,  Chicago,  1904. 

(i';)  l^rofessor  iMtXEST  W.  Browx,  of  Haverford  College  :  "  Mod- 
ern jNEethods  of  Treating  Dynamical  Problems,  and  in  Par- 
ticular the  Problem  of  Three  Jk)dies." 


ly.     The  ])()ston-  Collocjeiem,   190-"). 

(a)  I'rofessor  Henry  S.  AVhite,  of  Xorthwestern  University  : 
three  lectures  on  "  Ijinear  Systems  of  Curves  on  Algebraic 
Surfaces." 

(7;)  Professor  h"r>E])ERi<K  S.  A\'o()i)s,  of  the  Alassachusetts  Institute 
of  Technology  :  three  lectures  on  '' Forms  of  Xon-Kuclidcan 
S])ace." 

(c)    Professor  F]>WAi;i)  J>.  A'an  A'eeck,  of  A\'esleyan  T^niversity  ; 
six  lectures  on  "  Selected  Topics  in  the  Theoi'v  of  Divergent 
Series  and  Continued  I^'ractions." 
This  collo(piium  is  here  ])ublished  in  full. 

At  Commencement,  19()."],  Professor  dohn  Moni'dc  Van  AHeck. 
M.A.,  PL.]).,  completed  his  fiftieth  year  of  service  at  \\  esleyan 
I'niversity,  and  retired  shortly  after  from  the  chair  of  Mathematics 


X  I'KEFAC'E. 

and  Astronomy.  All  three  of  the  speakers  at  the  Boston  Collo- 
quium were  former  students  of  his,  one  of  them  being  his  son 
and  colleague  in  the  dei)artment  of  mathematics.  It  is  fitting  that 
this  volume  of  lectures  held  at  that  Colloquium  be  inscril)ed  to  him. 

T}ir)MA.S  S,   FiSKK, 

William  F.  OsciooD, 

Coinmittef  on  Pah/icdfion. 


CONTENTS. 


Preface 


PAfiE. 

vii 


Linear  Systems  of  Curves  on  Algebraic  Surfaces 

By  Hp:nry  S.  White 

Cremona  transformations  and  the  geometry  on  a  curve.     Liaear 

series  of  point  sets        .........  1 

Rational  surfaces  defined         ........  4 

Linear  systems  of  curves  on  any  algebraic  surface.         ...  5 

Enriques's  theorem  on  the  two  definitions  of  linearity    ...  7 

Hyperelliptic  plane  curves,  two  kinds  of  linear  systems.          .         .  12 
Surfaces   whose   plane   sections   are   hyperelliptic,    Castelnuovo's 

theorem  on  their  rationality         .......  14 

Picard's  exact  linear  differentials  on  a  surface;  those  of  first  kind 

exist  only  on  singular  surfaces     .......  18 

Poiucare's  and  Berry's  special  quartic  surfaces        .         .         .         .25 

Humbert's  liyperelliptic  surfaces  of  sixth  order      ...  27 


Forms  of  Non-Euclidean  Space 


By  Frederick  S.  Woods 

1.  The  First  Two  Hypotheses 

2.  Definitions.        ...... 

3.  The  Third  Hypothesis       .... 

4.  The  Line-Element  ..... 
o.  Geometry  in  a  Kestricted  Portion  of  Space 
G.   The  Fourth  and  Fifth  Hypotheses    . 

7.  The  Extended  Coordinate  System 

8.  The  Auxiliary  Sj)ace  2      •         .         .         ■ 

xi 


32 

34 

37 

. 

39 

45 

51 

52 

57 

xii  CONTENTS. 

9.   Forms  of  Space  which  aUow  Free  Motion  as  a  Whole       .         .     58 
Spaces  of  Zero  Curvature      .......     59 

Spaces  of  Constant  Negative  Curvature         .         .         .         .59 

Spaces  of  Constant  Positive  Curvature.         .         .         .         .60 

10.   Forms  of  Space  which  do  not  allow  Free  ^lotion  as  a  Whole   .      61 
Spaces  of  Zero  Curvature      .......     63 

Spaces  of  Constant  Positive  Curvature.         .  .         .65 

Clitrord's  Surface  of  Zero  Curvature      .  .  .  .  .69 

Spaces  of  Constant  Negative  Curvature         .  .         .         .71 

Selected   Topics   in  the   Theory  of   Divergent   Series   and  of 
Continued  Fractions 

By  Edward  B.  Van  Vleck 

Part  I.     Diverc^ent  Series. 

Page. 
Intioduction     ..........     75 

Lecture  1.   Asymptotic  Convergence         .         .         .         .         .77 
"      2.   The  Application  of  Integrals  to  Divergent  Series.     92 
"      8.   On  the  Determination  of  the  Singularities  of  Func- 
tions defined  by  Power  Series      ....   107 
"      4.   On  Series  of  Polynomials  and  of  Rational  Fractious,   120 

Part  II.     Al(;ebraic  Continued  Fractions. 

Lecture  5.    Pade's  Table  of  Approximants  and  its  Applications.    134 

■'      f).   The  Oeneralization  of  the  Continued  Fraction         .   154 

Bibliograjjhy     .  .  .  .  .  .  .  .  .  .    1()7 


LIXKAR     SYSTEMS    OF     CIMJVKS     OX     ALCiKJJllAIC 

SURFACKS. 

I'.v   !iKXi!V  s.  wiiiri:. 

CuAPiiii;    1. 

Trdns'ifion  jrniii    Plmw  ('trrvc.-<   to   Snrfdius. 

TnK  notion  of  ecjuivalence  as  foniinlated  in  i)roj(.'ctivo  jj;ooint'try 
has  siniplitieJ  givatly  the  study  of  algebraic  curves  and  surfaces, 
particuhirlv  tho-^e  of  low  order.  The  next  stej)  toward  a  wider 
survey  is  tlie  admission  of  all  hirational  transformations  of  the 
plane,  or  of  space  of  three  or  more  dimensions.  \\\  the  ])lane,  the 
theory  of  Cremona  transformations  is  no  longer  new,  and  the 
elements  are  familiar  to  all  students  of  geometry.  Xot  so,  how- 
ever, in  space  of  more  than  two  dimensions  ;  ])ro!)al)ly  for  the 
reason  that  nothing  is  known  atialogous  to  the  theorem  that  a 
plane  (Jr(>mona  transformation  is  resolvable  into  a  succession  of 
quadric  transformations  and  collineations.  And  even  in  ])lane 
geometrv  the  intricacies  of  the  transformations  themselves  Jiave 
kept  most  students  from  the  matter  of  higher  importance,  the 
jM'opei'ties  of  figures  that  remain  invariant  under  all  ti'ansforma- 
tions  of  the  group.  Yet  there  does  exist  a  bodv  of  doctrine  under 
th(>  accei)te(l  title  of  '•  (icometry  on  an  algebraic  cui'vc/'  and  a 
fair  beginning  lias  been  made  uj)()n  a  similar  theory,  the  "  Cieometi'v 
on   an   algebraic  surface."  ''      These   titles   ar(>   intended   to   cover 

*('(insult,  for  an  dutline  of  the  ncoiu.Hi'y  upon  an  alueliraic  ciirN-e,  PasraTs 
Rfjirrloriimi  '/./•  luilu'rni  M'ltlti'iiKilih.  I'art  11,  CliaptiT  \',  \\\  or  the  iiKire 
extended  artieles  :  (,'.  St'^i-e,  '"  1  iitroduzioiie  alia  L;t'(inielria  sopra  iin  ente 
al<i;et)riei)  -einplieeinenle  iiitinito";  Vl.  liertini,  "La  j^emiietria  delk'  siTie 
lincari  sopra  iina  eiirva  plana  seeondo  il  inetodo  aljjCehrico."  — i)()th  in  Aniidii 
ill  Mdti-iiHitira,  >er.  1',  vel.  "_!■_'  f  1  SUl ).  l'\)r  the  eurrespondini,'-  thedries  reuard- 
inir  sni'face>.  the  \>e>{  reA'renet'  is  to  tlie  eompi'ehensi ve  >innniary  liy  Castel- 
niinvo  and  Knriipies  :  ''Snr  i|ueh|Ues  recents  i-esnitats  (hins  hi  tlie<irii>  di-s 
surfaiH's  aln(^lirii|ne>,'"  Mulli.  Ainmhn,  vol.  1"-^  'ISUi'ii.  Supplementary  resuli^ 
are  suiinaari/.ed  in  a  later  ])apcr  hy  the  same  antlior<:  ''Sopra  aleuue  ipiestioiii 
fondamcntali  nella  teoria  di'lle  superlieie  alu^ehriehe,"  Aiuki/I  i(i  ,n<ilciiuitiv  i  jnim 
,■(/  iiii/i/inil,!,  sm-.  .">.  V(d.  t'.  (1!H)1  ). 

1  1 


2  Tin-:  i'.<)ST()X  coLT.ofirn^^r. 

only  such  ))i'()])('rti('s  of  a  (Mirvo  or  surface  as  appertain  to  the 
entire  class  ol"  curves  or  surfaces  that  can  be  related  birationallv 
to  the  fundanienlal  Ibi'iu. 

A  i)laue  alu'chraic  curve  may  have  its  order  chanu'cd  bv  a  Cre- 
mona ti'an-formation,  but  not  its  deficiency  (genrt',  Geschlecht). 
As  to  set>^  of  jxiiuts  on  the  cui've,  two  sets  whicli  to(2:ether  make 
u])  a  complete  intersection  of  a  second  curve  with  tlie  first  do  not 
lose  that  ])ro])erty  by  l)iratioiial  ti'ansformation,  if  we  exclude 
fi'om  consideration  fundamental  ])oints  introduced  by  tiie  transfor- 
mation itself."'  ]\rutually  residual  sets  of  ])oints,  and  corresidual 
sets,  ])reserve  their  relation.  Hence  the  group  of  sets  of  points 
corresidual  with  any  <i;iven  set  becomes  of  importance.  If  a  o^iven 
set  of  1)  points  lies  on  a  curve  of  deficiency  p,  and  if  a  corresidual 
set  can  l)e  found  containing/;  arl)itrary  ])oints,  then  these  numbers 
are  connected  l)y  the  relation  constituting  the  Kiemann-lioch 
theorem 

/■  =   />  _  y,  4-   o, 

whei'e  II  Is  zero  if  Z-*  ^  'Ip  —  "2. 

The  totality  of  all  sets  of  7)  points  corresidual  to  any  one  set  is 
termed  a  group  or  i^i'iues,  and  is  denoted  by  a  symbol  ///;.  Such  a 
series  i<  called  cninitlitc.  If  by  any  algebraic  restrictions  a  series 
is  sepai'ated  oul  from  it,  of  coui'se  that  would  be  called  incom])lete 
or  partial.  For  example,  on  a  ])lane  nodal  cubic  a  series  //:  is 
cut  out  bv  all  straight  lines,  incoin])lete  IxM-ause  anv  three  ai'bi- 
ti'ai'v  ))oints  of  the  curve  are  cori'csidual  to  any  otliei*  three. 
l''\'ei'v  scries  i/'j,  can  be  cut  out  upon  the  fundamental  curve  bv  a 
lin(>ar  sv«tem  of  auxiliary  curves  whose  c(|uation  may  be  written, 
with  /•  pai'ametci's  : 

/•:-+  '^|■\  +  'J\+  ■■■  +  'ij-',-^^- 

As  (111  a  <inglc  curve  sets  of  j)oiuts,  -o  in  a  plaiu',  linear  systems 
of  cui-xc-  arc  studied.       !>\'  cvcrv  biratioual  transfoi-mation,  linear 

(  >i    it   \'.c  ciiiiiliiv    iiu   :iii  \  i  1  ill  ly  rllivcN  except  siieli  :i-~  ;in>  :ii!j(iint  to   that  cdii- 
tainiii^'  t  III    I'l'iiit  -It-. 


t.ixi;a]{  svs'ri:.Ms  of  (  ri:\'i:s.  .'i 

systems  iirc  carried  over  iiitu  liiieai"  s\steni.s.  A  complete  linear 
svstem  is  detined  most  easilv  hy  sj)e('ifvin^  tlie  multiplicity  that  a 
('iirv(>  (if  the  svstem  must  have  in  each  itoint  of  a  fundamental  set 
ami  1)V  pi'e<crii)ing  the  n, ■(!<■,■  ot'  the  curves.  Thus  (",■,':;;)  can  in- 
dicate that  in  ^/^  every  curve  is  to  have  a  multii)le  ])oint  of  order 
at  least  -^-i.  etc.  If  the  base  ])oints  alone,  with  their  respective 
multijdicities,  determine  a  system  under  consideration,  that  system 
is  termed  cinnplcte.  If  the  base  ])oints  actually  impose,  for  curves 
of  ordei'  //(,  fewer  conditions  than  would  he  expected  from  their 
scvei'al  multi])licities,  the  system  \^  special ;  otherwise  it  is  rcf/n- 
/(ir.  It  is  an  important  theorem  that  no  set  of  r  base  })oints  can 
be  so  /nrtifrd  as  to  jiroduce  an  (/■  +  l)th  variable  multiple  point 
on  the  curves  of  the  system  ;  /.  c,  the  nndtijde  })oints  of  the 
generic  curve  of  a  plane  linear  system  lie  all  in  the  base  points  of 
the  systeui. 

Afljoiiif  curves  of  a  linear  system  are  familiar  to  the  student 
of  function  tlieory  ;  they  have  in  every  nudtiple  point  of  order  .s 
for  the  iriven  system  a  multiplicity  of  order  at  least  -s' —  1.  The 
adjoints  of  order  lower  by  •">  than  the  original  system  are  im})ortant 
from  the  fact  that  tliey  transform  always  into  the  corresponding 
system  of  adjoints  to  the  transformed  curves.  On  this  account 
the  term  adjoint,  as  used  ordinarily,  imj)lies  a  curve  of  order 
/;/  —  '■]  unless  differently  specified.  Second  (idjoiitts  are  adjoint  to 
adjoints  of  the  system,  etc.  The  employment  of  successive  adjoint 
sy.-^tems  as  a  means  of  investigation  is  due  to  S.  Kantor  and  to  G. 
(  astelnnovo,  the  latter  acknowledging  the  priority  of  the  former.* 
On  every  ('urve  its  adjoints  cut  out  a  unique  com])lete  series  f/i^^l^^ 
called  the  i-diionlcd/  srries.  The  deficiencv  of  the  first  or  second 
adjoints  of  a  linear  system  is  denoted  by  J\  or  /■*.„  and  may  be 
termed  first,  or  second,  canonical  deficiency.  Aside  from  the 
canonical  series  u})on  curves  of  a  system,  the  most  important  are 
llic  cliffrdcferlsfic  series  of  the  system,  that  is  the  totality  of  sets  of 
points  in  which  two  curves  of  the  system  inters(H't.  \i'  a  ])lane 
linear  svstem   is  complete^,   then   the  ehfiraderistic  srries  on   each 

■■See  .Until.  .[nn<'l>'ii,  vol.  41  (1S',)4),  p.   TJ?. 


4  Tin:  ]i()s')-()X  coLLocH'ii'M. 

curve  is  a  coniphic  se/-ie.-<  uj)oii  that  curve.  So  far  the  definitions 
and  propositions  refer  to  curves  in  a  })huie  ;  the  question  is  in 
order  whether  tliey  can  he  transferred  to  systems  of  curves  lying 
U])on  curved  surfaces. 

First,  it  is  noticed  that  by  means  oi"  a  linear  system  of  curves 
the  ])lane  may  be  related  ])oint  for  point  to  a  surface  in  space  of 
three  or  more  dimensions.''  If  the  system  is  ^--fold  infinite,  k  -\-  2 
members  of  the  system  can  be  related  arbitrarily  to  A'  +  2  hyj^er- 
])lanes  in  space  of  /.--dimensions.  Take  J:  =  3  for  ease  ;  then  a 
curve  of  the  system 

may  be  assiuned  to  a  j)lane  [n^:  n.,:  i'.,/.  n^)  in  ordinary  space. 
Curves  throutrh  one  point  become  then  planes  through  one  point, 
and  the  oc'  ])oints  of  the  j)lane  become  the  oc'-  points  of  some 
alu'cbraic  surfiu-e  f.  All  such  surfiices  are  called  rdiiomtl.  Simi- 
larlv  a  linear  familv  of  curves  tri})ly  infinite  u])on  any  surface 
n'late  that  surface  i)oint  for  ])oint  to  another  surfiice  in  threefi)ld 
space,  linear  svstems  of  curves  in  one  givinu'  rise  to  linear  systems 
Uj)on  theothei",  and  f/w  /j-misfonurd  ,sv/.s7r///  //■///  /(fch  j'u iiddiiuiifd/  or 
//(ISC  l)f)!)ifs.  The  value  of  such  projeeti\'(dy  related  ])ictures  of  a 
linear  svstem  of  curves  was  fii'st  emj)hasized  bv  (".  Score. 

Secondly,  thn-e  are  surfaces  not  rational.  I'^tr  exam])le,  there 
ai'c  iri'ational  iMded  surfaces.  IJut  for  many  purposes,  rulcfl  sur- 
faces and  rali(tnal  surfac(s  are  classed  (ogether  and  constitute,  with 
their  eijui vali'iits,  a  small.  iudecMl  an  e\cej)tional,  cla-s  in  the  vast 
field  of  alu'cbi-aic  >ui'faces.  I'laiies  ai'c  al<o  rfyu/nr  surfacis^  that 
i-,  thcv  lia\c  their  gcoineti'ical  and  luiinei'ical  (oi-  arithmetical) 
defieieiicie-  e(|Ual,  as  will  be  explained  dii'ectlv.  On  t'egulai'  sur- 
faces, nio-t  of  the  theorem-  upon  linear  svstem-  of  curve.-  on 
i'atioii;d  -ui'lMee-  I'etaiu  tlicir  \-ali(litv:  not  -o  on  the  ii'i'egidar. 
New  chai-aeteiv-  crop  out  in  the  -\-- ten  is  of  curve-,  eliaracters  wliicli 
iiidieate  the  nature  of  the  -uii;ie<'.       \\\\\  the  liiiearit\-  of  a  svstem 

'■    lvM-r|iI  ininl   cil-i's  ;i  re  il  i-.russril    li\-    tju' ii  UK '^  :     "   luccrcllr  lii    LI'ci  Ullfl  I'i.l    Suilt' 


TJNKAlf    SYSTKMS    OF    Cl'KVKS.  ■) 

of  curves  is  still  susceptible  of  precise  d(!finition,  and  tliat  in  two 
ways  whose  etjuivalence  constitutes  an  important  theorem. 

Jt"  on  any  surface,  rational  or  not,  there  exists  a  system  of 
curves  d<)ul)lv  infinite,  such  that  two  arbitrary  points  determine 
one  and  onlv  one  curve  containing  tliem,  that  may  be  termed  a 
linear  net  upon  the  surface  in  question  ;  and  Enriques  proves  that 
the  cc"  curves  of  such  a  system  can  bo  jjrojcct'irchi  related  to  the 
straight  lines  of  a  ])lane.  If  the  series  is  oc',  and  if  three  arbi- 
trary ])oints  determine  unicpiely  a  curve  of  the  system  which  shall 
contain  them,  then  its  curves  are  referable  projectively  to  the 
planes  of  three-s})ace,  etc.  Only  .^inqtfi/  infinite  systems  escape 
this  fu'-reaching  theorem,  and  thus  give  rise  to  a  most  interesting 
unsettled  ({uestion,  indicated  by  Castelnuovo.* 

Definitions  of  resi(b.ial  and  corresidual  curves  upon  a  surface 
are  those  which  any  one  could  formulate  at  once  from  the  use  of 
these  tei-ms  for  sets  of  ])oints  upon  a  curve;  their  significance 
uj)on  a  twisted  curve  is  the  same  as  Uj)on  its  ])lane  i)rojection. 
So  of  conq)letc  systems,  both  of  curves  and  of  surfaces,  the  latter 
admitting  of  course  multi})le  cui'ves  as  well  as  base  j)oints.  For 
a  surfiice  of  order  m,  tlie  adjoints  invariantively  related  are  of 
order  ui  —  4,  containing  as  (-s  —  l)-fi)]d  curve  every  x-fold  curve 
of  the  given  surface.  If  these  first  adjoint  surfaces  form  a  /--fold 
infinite  linear  system,  the  number  k  is  an  invariant  of  the  surface 
and  is  termed  its  (jeomcfric  defirioici/  (jj^).  Attempting  to  express 
this  number  in  tei'ms  of  the  order  //(  of  the  surface,  the  order  d 
and  deficiency  -  of  its  double  curve  (if  any),  and  of  the  number 
/  of  trij)]e  ])oints  on  this  double  curve,  one  would  find  a  second 
number 

y,^^  =--  IJni  —  1)  (,n  —  2)  (ui  —  :V)  —  il  („i  —  4)  -f  2/  +  -  —  ], 

called  the  unincricdl  (Icjicii'Hoij  of  the  sui'fiuH'.      This  number  also 
is  an   invariant  of  the  surface,  as  Xoctiier  first  ])roved,  and  may 

■■  (':istelnui)V()  :  "  Alcuiii  risultati  siii  sistemi  liin'iiri  di  curve  appiirteneiui  ad 
uiKi  superlicie  alt^chrii/a."  Miuwrif  di  vidii'imdicd  e  di  ilsicu  delhi  Soriiia  Italimin 
dell,'  S'-ip-nz,',  ser.  ?>,  vol.  10  (LS'.Xi),  \)\).  82-10-2.  See  especially  the  close  of  his 
{ircface. 


6  THE    BOSTON    COLLOQUTLDI. 

be  lifher  ('(jUAtl  to  or  /ch-s  iluin  y;  ,  but  nev(>r  greater.  Ivatioiial 
surfaees  have  j>^  =  />^  =  0  ;  ruled  surfaces  have  p^^  negative.  If 
y>  =y>„,  then  the  above-mentioned  theorem  of  P^nriques  eoncorn- 
ing  linearity  holds  true  also  for  systems  which  are  only  simply 
infinite.  Surfaces  of  the  first  adjoint  system  cut  out  upon  a  given 
surface  a  system  of  curves,  each  of  deficiency  7;"  or  less.  This 
invariant  number  y> '-  we  may  call  the  cdiiorucdl  dcjiciency  of  the 
surface  ;  the  curves  form  an  unique  com]>lete  linear  system,  just 
as  do  the  point  sets  of  the  canonical  series  on  a  })lanc  curve. 

The  definitions  here  given  are  but  a  part  of  those  found  useful 
in  this  fascinating  branch  of  geometry.  The  true  wav  to  learn 
something  of  the  subject  is  not  to  master  first  all  its  definitions 
and  distinctions,  but  to  study  the  proofs  of  some  few  leading 
theorems.  Such  are  Enri(jues's  proof  of  the  equivalence  of  two 
geometrical  definitions  of  the  linearity  of  a  system  (mentioned 
above),  and  the  following  less  elementary  propositions  : 

1.  Sui'faces  whose  jilane  or  hyperplane  sections  are  irreducible 
unicursal  curves  are  either  ruled  or  rational  (Xoether).'^' 

'1.  So  also  surfaces  whose  ])lane  or  hyperjilane  sections  are  irre- 
ducible elli])tic  curves  (Castelnuovo),T  or  hypcrelliptic  of  any  de- 
ficiency 77  ( lMiri(jues).:j;  l''or  plane  sections,  not  hy])erelli])tic,  of 
deficiency  -  >  2,  the  corres))oiiding  theorem  is  not  yet  fully 
known. ^J      The  proof  of  this  theorem  I  shall  give  in  full. 

8.  l'])(in  any  algebraic  surfacx^  ./(•'"?  !h  '~i  0  ~  **  '^  linear  diflfer- 
ential  of  iii'st  kind  is  said  to  exist  (l^icard),  if  an  ex])ression  in- 
volving foui- I'ational  functions  l\,  /-*.„   /"*,,,  /',,  of  the  co("'(rdinates  : 

is  finite  and   determinate,  in(l('])eiident    of  the  path  of  intcgi'ation, 

*  Ndct  licr's  tlu-iiriMu  i<  iikh'c  u'ciu'fal.  Sec  Muth.  Ammlin,  mA.  I!  (  ls71)  : 
''  I  Clicr  I'liiclii-n,  welclie  Scliaarcii  I'at  i(  niak-r  (  ui'vcii  liesitzcii." 

i  ■' Siillc  >ii])rrticic  al^cliriclic,"  i-tc,    Liiir,-i  Uriiiliciiiifi,  .Jaiiiiarv.    IS'.tl. 

;   ''Sui  -i-temi  lincaii."   etc.,   Mulii.    .liniii/,ii.   viil.  -lH  I  IS'.i:)),   pp.    17it    I'.'lt. 

i  t"(ir  full  inruriiiat  ii)ii.  sec  tin'  sccdiKJ  ])apcr,  ritcil  ahnxc,  of  (  aslcliiiiovo  ami 
I'airiipii'rt.  I  rcLTrt  that  tlii--  paprr  had  not  ci  iinr  ti>  iiiv  not  ice  iK'fnrc  LciTiiiL'' these 
lecture-. 


ijxi:ak  svstk.ms  of  ctkvi'.s.  t 

when  taken  upon  the  siiriacc  ix'twecn  any  two  arl)itrai"y  ])oints. 
If"  the  -surface  _/'=  0  is  a  co:if,  sucli  dilffrentials  exist,  ior  they 
are  tlu'  al)elian  differentials  ot"  first  kind  Ujxtn  its  plane  sections. 
I^ieard  proves  ^^'  that  if"  the  surface  _/'=  (»  have  uo  inultipU'  points 
or  curves,  then  no  such  differential  can  exist  uj)on  it.  There  are 
ho\\'ev(n'  surfaces  of  all  orders  above  tlie  third  which  contain  (or 
admit)  one  such  inteural  ;  others,  from  the  sixth  order  u[)war(l, 
which  admit  two,  and  so  on.  These  surfaces  and  tlie  mode  of  dis- 
coverino-  them  and  of  detininu'  them  have  been  the  occasion  of 
some  of  the  most  interesting  studies  of  ricdrd  and  Ibimbert.  The 
elementary  ])art  of  Ticard's  first  jiajier  upon  this  topic  T  shall  give 
in  some  detail,  indicating  in  conclusion  certain  j)oints  that  might 
prove  worthv  of  further  studv. 

C'irAI'TKR    2. 

Liiifdr    Si/sfeiii^  of    ('ii/Ti'fi    Oil    (III   AI(/eI)r(J.tc   Surface.      The    Tu-o 
( ''eoiiiefr/e   J)efiii!fion.^  are    Cuncorddid. 
Is  plane  geometry  a  linear  svstem  of  algebraic  curves  is  defined 
analytically  by  an  e(juation  containing  linearly  and  homogeneously 
two  ov  more  pai'ameters  ;  as  for  example  : 

X,0„  +  \,0,  +  \<^,,  -f  . . .  +X>^  =  0, 

the  \,'s  being  parameters,  and  the  (i"s  a  set  of  polvnomials  homo- 
geneous of  lii-;e  degree  in  the  curi'ent  coiu'dinates.  This  is  called 
a  «--fbld  infinite  (oc")  linear  system.  .\s  we  restrict  our  field  to 
include  only  systems  delined  l^y  fixed  base  ]K)ints,  the  curves 
(^.  =  0  must  be  sui)posed  all  to  contain  the  base  points  of  the 
system.  In  a  plane  such  a  system  may  l)e  studied  bv  means  of 
its  ('(juation,  but  for  oilier  surfaces  one  must  either  assume  an 
analytic  representation  as  definition,  or  else  take  such  geometric 
features  of  a  plan(>  linear  sy>tem  as  seem  most  important  and 
tran-fcr  them  to  sets  of  curves  on  surfaces  in  genei'al.       \\ C  follow 

■*  I'ii'ill'i.l  et    Siinai't  :     Thoiric    <l<y   fitilctiniiH    <ihi('l>i  'kiU'H  dr    ilrii.r    va  ,-inl,h  .<    indrji/'n- 
(l'(iil''.<.  vol.  1    (1S;»7),  \y\K   ll',l"l-J(). 


<s  Tin-:  B()s'ix)x  (X)LLO(a^n'^^r. 

the  latter  j)lan,  and   two  defiiiitions  naturallv  jn'csent   tlienisclves. 

I'^irst,  iisinti"  an  anxiliary  hy])er-s])a('('  of  as  many  dimensions  as 
the  system  ofeurves  exhibits,  an  oo''  system  of"  curves  on  an  alge- 
hi-aie  surface  is  (tailed  linear  if  its  elements  (the  individual  curves) 
can  l)e  put  in  corres])ondence  one-to-one,  ])rojectively,  with  the 
livj>er-planes  of  a  S])ace  of/'  dimensions,  ,V. 

Second,  usinn-  no  auxiliary  outside  the  points  of  the  surface 
itself,  an  oc'  system  of  curves  on  an  al_<i:;ebraic  surface  is  called 
linear  if  through  /•  o'cneric  ])oints  of  that  surface  there  j)asses  one 
and  only  one  curve  of  the  system.  Tiiis  definition  is  to  be  used 
onlv  v.lien  /■  7^  1 .  ^^)r  if  /■  =  1 ,  the  <:;enerat()rs  of  a  ruled  sur- 
face would  fall  undci'  this  definition,  and  one  sees  immediately 
the  imiir'oprictv  of  callinu'  them  a  linear  system. 

Notice  that  a  system  linear  under  the  first  definition  must  also 
be  linear  under  the  second.  For  by  i-elating  curves  to  hyper- 
planes  we  I'clate  the  tdgebraic  surface  7*^  to  a  new  surface  /"  in 
N,  as  explained  in  the  ]ireceding  chaj)ter  ;  and  through  /•  j)oints 
on  /•"  thei'c  will  pass  one  liyi)er|)lane,  hence  thron<i;]i  /•  ])oints  in 
/•'  there  will  ])ass  one  curve  of  the  svstein  and  no  more.  The 
fii'st  defmition  thei'cfore  includes  the  second  ;  does  the  second 
include  the  first?  ^\'e  shall  show  that  it  does,  so  that  the  two 
definition^  shall  be  j)i'oven  ecpiivalent  fbi- all  cases  except  r  ^  1, 
that  is,  fbi'  all  excej)t  lineai'  sheaves  or  pencils.  The  proof  is 
essential Iv  that  of  l'jii'i(|Ues  ''''  as  })resent(Ml  by  Seo;re.T 

'fwo  Icnunas  mav  well  precede  the  theoi'cm. 

Ll•;^tM.\  f.  rrdjc'-t'iril II  (if  tiro  f(tfsj}(iiu.-<.  '\  \\o  flat  sj)aees  of // 
diiiien-ioii~,  >'  and  N  ,  can  be  |)i'ojectivelv  related  bv  assiniiinii' 
In  aiiN'  /'  4-  -  ueneiMc  hvjx'i'planes  or  N  ^'s  of  the  first  anv  //  -|-  2 
Li'ciieiMc  >'  I's  of  the  second,  (Hie  to  one,  as  corres])on(lin!i'  lorms. 
'fhc  pi'oof  i<  b\-  mathematical  induction  ;  to  i^ain  a  cleai'  idea  i>f  it, 
-late  it  \'nv  points  iii-tead  of  hxperplanes,  and  model  the  transition 
iViiiii   >>    and    .s     to  >■     I  and    N '    ,   upon    \'on    Staudt's  I    transition 


'  Ijii  ii|iii-  :    "I   n.i  (|ih'-ii(iiic  siilhi    1  iiic:i  rit:~i  iln  si^ti'iiii  di  ciii-vc  :i|  i]i:i  rtciicnt  i 

;i(l    un:i    -lllirl-liric  .■ili^i-l.i-i.'.-i."         KmIiic,     /,(//r,  /    /,'-/<'//■■-///(.    .Illlv,    IS'.i:'.. 
■  M-i:iv:    l,.r.   ri,   in   ,| //  ,';    1/.,/,  ,„,,//.<,,  m-I'.   'l.   v,,l.   l!:^,   ^  liT. 


LIXEAIJ    ►^VS'l'K.MS    OF    CUia'KS. 


9 


from  >'„  to  Nj.  Ill  N,  take  any  •")  points  A,  II,  ( ',  ]>,  A',  such  that 
no  four  lie  in  a  })htiu',  aiul  in  N.'  simihirly  .1',  II  ,  C ,  D' ,  A",  like 
K'ttcrs  (U-notiny-  (•orivs])()n(lino;  j)oints.  In  the  plane  C 1  >' K'  uv 
>'.,  call  ]^  the  j)oint  of  intersection  with  the  line  .1 ' !>  ,  and  in  <  '1)K 
or  >'.,  let  I'  i)e  on  the  line  .17).  As  7*  and  /''  must  correspond, 
this  gives  4  jtoints  in  >'„  corresponding'  to  4  fixed  ])oints  of  >'.',  and 
therefore  hy  hypothesis  fixes  the  j)rojectivity  between  the  two 
plane-.  44u'  ])eneil  of  ravs  in  N,  through  P  corresponds  projec- 
tively  to  that  through  P'  in  the  other  ]ilane,  >'.',.  If  now  (^> 
denotes  any  point  of  >'..,  to  find  its  corresponding  ])oint  ^/  in  N.^ 
let  (^  he  jtrojected  from  .1  and  from  11  into  two  })oints  .1,  and  II^ 
of  the  {)lane  >'.,.  'Jdiese  are  collinear  with  1\  and  we  can  find  their 
corresponding  points  .l^  and  I)\  collinear  with  P'  in  S\^  and  so, 
by  using  .1  and  P'  as  centers  of  ])roiection,  the  point  ''/  desired. 
Points  on  the  line  AP  itself  have  their  corres])onding  j)oints  fixed 
by  the  assignment  of  ;>  ])oints  .1,  P,  P  to  the  points  A' ,  B  ,  J' 
respectively  in  the  line  A'  P' . 

IvKMMA  '1.  In  an  zc'-  algebraic  system  of  irreducible  curves 
U])on  an  algebraic  surface,  if  tlu'  svstem  is  linear  according  to  the 
second  definitit)!!,  tlien  the  points  (jf  the  surface  form  sets  o\'  n  (some 
finite  number),  such  that  if  a  curve  of  the  system  contains  one 
])oint  of"  any  set  it  must  necessarily  contain  also  fhe  other  n  —  1 
points  of  that  -ei . 

The  ])roof  rest<  on  tlie  algebraic  characters  of  the  svstem.  Call 
the  system  {(')  and  anv  curve  of  the  svstem  ( \.  Select  anv  point 
.1|  of  the  surface.  It  does  not  determine  a  curve.  Let  C,  and  <  [, 
l)e  any  two  irreducible  curves  through  A  ^.  Ihev  intersect  in 
n  —  1  other  ])oiiits  /{.,,  .1,,  •  ■  •,  .1  ,  {ii^-\).  Since  two  of  these 
l)()i!its,  './/.,  .![  and  .1.,  lie  on  two  curves,  they  must  lie  on  an 
infinity  of  curves  ;  /.  c,  it  will  re(|uireat  least  one  additional  jxunt 
to  determine  a  single  curve  from  among  those  that  contain  both  .1, 
and  A.,.  Jt'  P  is  a  generic  j)oint  not  on  all  curves  that  contain  J,, 
tlu'u  l)y  hypothesis  the  two  points  .1^  and  P  detcu'inine  one  curve, 
which  shall  l)e  denoted  by  ('..  .\  Uo  among  the  curves  that  con- 
tain A^  and  .1..,  at    least   one  will   contain    the  additional    point  /'. 


10  rilK    BOSTON    COLLOCa  H.M. 

This  can  he  iioiic  oilier  than  ( [.,  hence  the  curve  which  is  deter- 
mined l)v  .1,  and  the  ojeneric  point  /^  will  contain  also  .!„.  l^y 
parity  ol'  i-easonini;"  it  niust  contain  as  well  ^1.,,  •  •  •,  .1  .  Uut  as  P 
was  anv  ])()int,  ('.,  was  any  curve  through  .l,,  conseiiuently  every 
curve  ot'(C')  that  coiUains  an  arhitraiy  point  A^  must  contain  also 
//  —  1  other  determinate  points,  as  asserte<l  hy  the  lemma. 

The  j>rin(Mpal  theoi'em  can  now  he  jiroven  if  two  facts  are  estah- 
lished.  First  the  theoi'em  should  he  found  to  suhsist  i'oi"  the 
])articular  ease  i-='l,  so  that  the  hasc  may  he  j)rovided  to)'  a 
mathematical  induction.  Then,  secondly,  the  mode  of  induction 
employed  in  Lemnui  1  must  be  shown  to  he  ap})lical)le  to  a  sys- 
tcMU  of"  curves  conforming  to  the  second  definition. 

J^Ai;  TierLAi:  Tin:()i;E.M.  .1  doiih/j/  infijiife  (//;/rhroir  si/slrm  aj' 
i i-i-f(l ucihlc  (ihjchrii'n'  ciirrc-^  uj/on  (iiri/  ahjvbrdir  snffar,  caii  l>c 
hroiKjht  into  (I  Oiic-to-oiic  nhifioit  vlih  the  Hi/.sfci/i  of  oil  lim  ■<  in  o 
j)l((ii('  hji  (iKsi</iiii}(i  lo  four  (irhifrdi'ili/  c/ioscii  c/'/tc-s  (;f  tin'  -^//sft m 
(ho  i/ircc  t/iroi'(//i  one  poinf),  Jour  (irhlfrcrilj/  c/ioxcii  linrs  of  flu' 
jtloiic  {no  llirri'  tliroiKjIi  our  jjoiiif\  (tf<  coi-rcxiiondi nr/  lims,  oiid  hif 
rcijiiiriiHj  jnrflier  tliof  to  chitcs  Iiari/H/  (i  vonunoa  point  shall  nor- 
ri'sjxtnd  lines  irilh  a  point  in  coiiinioii. 

10  j)rove  this,  associate  (!very  set  of  ni  point^,  such  as  llie 
vl|,  .1.,,  •  •,  .1  ,  of  Lemma  '2,  together  as  one  element  .1.  Then 
there  is  upon  tlu' surface  an  x"  >y>tem  of  (''>  and  a  .-ecoud  ~ystem 
of  J's  I'elated  thus:  Two  genei'ic  C ''s  determine  one  A  aii<l  two 
J  V  determine  one  < '.  Now  these  are  ])recisely  t  he  inci<ienc<'  rela- 
tion- npon  which  de|)ends  the  familiar  ])i'oof  that  foui"  line-  of  one 
plane  and  four  of  another  deternnne  a  j)roiectivil v  of  the  two  sys- 
tem~  of  line-;  here  the  lines  and  point-^  ot"  the  one  jilane  are 
replaced  1)\-  element-  '' '  and  .1.  The  i'e(|ui,-ite  of  continuity  is 
|tro\-ided  fill-  !)v  the  li  v])ol  lie-i-  that  the  s\-stem  is  ot"  algehi'aie 
eliaraeter.  'I'herefnre  the  line-  of  a  plane  and  the  eiii'\-e-  oi'  the 
^\--!ein  {(')  -land  in  a  one  to  one  relation,  as  a--ei'te(l  hv  the 
theiii'eni.  Tlii-  relation  is  called  |)i'oiecti ve,  meanini^'  liiat  it  is 
intlepeiideiii  of  the  particular  foiii'  paii's.  line  and  cui"\"e,  that  may 
lie  -elected    to    determine    the    colTespondeUCC.       ()tliei'\vi-e    -tate(l: 


[JXKAR    SYSTEMS    Ol'    tTRVKS.  11 

Ii"tlie  lines  (if  two  ))lanes  arc  relatc'd  in  the  mode  above  (,]escril)C(l 
to  the  curN-es  of  a  svstem,  the  ])lanes  aiv  thereby  projectiN'cly 
related  to  each  other. 

As  to  the  second  matter,  it  is  needful  to  sliow  that  the  ch'nients 
iHe(l  as  auxiliaries  in  Lemma  1  have  unique  analoo^ues  in  a  sys- 
tem, triply  iuiiiiite,  of  curves  confoi'minsj  to  the  second  delinition. 
A\'hat  were  called  jioints  there  liave  become  curves  here,  hence 
the  lines  and  planes  must  be  replaced  by  oc'  and  oc"  systems 
of  curves.  AVe  need  only  examine,  accordingly,  whether  the 
jxistulate  :  a  line  and  a  plane  intei'sect  in  one  point,  retains  its 
validity.  Let  a  "line"  be  given  by  two  curves,  a  "  ])lane  "  by 
three  ;  or  to  adhere  more  closely  to  the  definition,  consider  an  N, 
given  by  two  points,  a  and  h,  and  an  S.,  consisting  of  all  the 
curves  of  the  oc*  system  >',  that  ])ass  through  a  third  point  c. 
'Jdien  will  >',  and  N,  have  in  common  one  and  only  one  curve. 
l'^)r  in  the  >'„  tliere  is  an  -S.'  containing  the  point  a.  ;  in  this  S'^ 
there  is  one  curve  (J  that  contains  the  jioints  h  and  c  (and  by  the 
explanations  of  the  above  theorem  we  see  that  it  must  contain  all 
the  intersections  of  any  two  curves  fixing  the  >S'J.  As  containing 
'/  and  h  it  lies  in  N,  ;  as  containing  c  it  lies  in  S.,,  and  as  contain- 
ing  these  three  arbitrary  ])oints  it  is  by  the  definition  unique. 
d'hei'efore,  all  the  constructions  of  Lemma  1  have  theii'  uni(pie 
anal(»gues  in  the  svstem  >'.,. 

\\'(!  conclude  that  the  ti'ansition  from  an  x"  system  to  one  oc' 
is  j)ossible,  and  that  for  /■  =  .">  the  first  and  second  definitions 
ai'e  e(|uivalent.  MufcJi-s  unild nd'ts,  the  induction  from  r  =  ?n  to 
/■  =  //,  4-  I  can  be  made  by  similar  means.  Uecapitulating  we 
have  thercibre  the  theorem  : 

^\  ii  zc'  <il(/(hr<ih'  .^ii.-<if_'iii  <>J  irrcihifUilc  (ih/chrdlc  riirrc.^  iipon  an  if 
(ihjdtra'n-  siirjdcc  ix  /lucnr  if  cUlier  (/)  its  ('/ciiuiif-s  ran  he  jinf  in  a 
otK-ln-Due  cor/'(.^'j)Oii(l<'iii-c,  jjroj'rc/irc/i/,  irifh  tlic  hijjx'rpldius  of  an 
r-J(ihl  sjiftcc  ;  or  (,')  ij'  //n'oio//i  r  (/cncric  jtoinfs  of  the  siirfdcr  ihcre 
jidsscs  one  dud  onl if  one  cui-rc  (,f  the  xiistcui.  For  r  >-  /  tlie.^e  firo 
(Ir/iiiliif/  jji'Oiir/'fics  cdii  Jtr  inf'r/'fd,  cdcJi  from  (he  other. 


1-  TiiK  BOSTON  ("oij/)Quir:\r. 

ChAI'TER    o. 

Siii-j'dcrs  irho-sc   Plane  Sfcfions  are  IL/pci-ff/ipfic   CV/'(v'.v. 

Pj.axk  curves  of  anv  deficiency  above  1  may  l^e  hyperelliptic, 
and  those  of  deficiency  2  are  necessarily  so.  Tlie  specific  feature 
of  an  livj)erelliptic  ])lane  curve  of  order  //  is  this,  that  its  adj((int 
curve-  of  order  /;  —  •"'),  its  "  </)-curves,"  arrange  its  points  in  ])airs. 
That  is,  if  a  (/)-curve  contains  any  one  point  /■*  of  a  ]iyj)erelli]>tic 
curve  (',  it  will  of  necessity  contain  a  second  determinate  ])oint 
Q  of  (';  then  /•*  and  Q  form  what  is  called  a  coiijnj/dfc  pair  ;  each 
is  the  (■niijiKjdte  jioinf  of  the  other.  It  is  well  known  that  a  (f)- 
curve  can  he  found  which  shall  contain  y>  —  1  arbitrary  points 
of  (',  where  jt  denotes  the  deficiency  of  the  curve  ('.  These  facts 
lead  to  interesting  conclusions  about  any  linear  system  of  hyjier- 
elliptic  curves  in  a  ])lane,  or  in  any  rational  surface. 

In  a  plane,  a  linear  system  of  liyperellij)tic  curves  may  be  of 
tlie  first  iir  second  kind.  Jn  a  system  of  the  first  kind,  a  curve 
jias.-ing  thi'ough  any  one  point  is  not  tliereby  necessitated  to  })ass 
through  a  determinate  second  ])()int  ;  in  a  system  of  the  second 
kind  this  compulsion  does  exist,  and  all  curves  of  the  system  that 
cuntain  a  point  //"  contain  also  (^,  its  conjugate  j^oint.  Of  the 
sc<'()nd  kind,  fbi-  exam])le,  is  a  certain  family  of  plane  sextics  hav- 
ing double  ])()ints  in  seven  common  points  of  three  cubics  :  c^^  =  0, 
(}) ^  =  (I,  (p   =  (I.      The  ecpiation 

gi\-c-  a  bucar  s\"steni  of  sextic>,  the  ^ '^^.  being  arbitrary.  ( )iitsi(le 
of  the  -e\'eii  base  points,  let  an\'  j)oint    /'  be  on  both  cubics  : 

(/),  =  O         and         (/).,  =  (I. 

'Iheii'  iiiiiih  intei'-<'elioii,  O,  is  determined  b\-  the  eighth,  a  familiar 
theoi-em  :  and  .-e\tie>  of  the  s\-<tem  which  jtass  through  /',  being 
gl\iil   b\-  llie  eijUatiou  ( aeeor<l  i  iig  to  Xoether's  theoi-ein) 

mu-t    eoiiiaiii    ai-o   the    remaining    intersection    O  of  c})^=^{)   and 


jjm:ai{  sys'ii::\is  of  ('i'itvi:s.  i-> 

(^6,  =  0.  Xotice  that  (f)^,  (f).„  c^.are  adjoint  (/)'s  of'all  scxtics  <>f"  the 
system,  so  tliat  </  is  tlie  conjugate  })oint  of  I'  on  every  sextie  tliat 
eontains  theni. 

We  mention  systems  of  this  s(M'()n(l  kind,  only  in  order  to  cx- 
olnde  tliem  from  iurther  discussion  here.  Let  (//)  l)e  an  dc"  system 
of  the  first  kind  of  hyperellij)tie  i)hine  curves  //,,  Jf.,,  etc.,  of  order 
//,  and  let  ((f))  he  the  system  of  adjoint  curves  of  order  //  —  ."'),  /.  r., 
let  the  curves  (/>,,  (/>„,  •  •  ■  have  as  (/"  —  1  )-fold  points  the  /-fold  base 
points  of  the  system  (//).  Consider  any  point  7"*  of  the  })lane. 
Its  conjugate  Q  on  anv  curve  If  of  the  system  must  lie,  by  defini- 
tion, uj)()n  every  (/)-curve  containing  1\  Since  Q  is  a  variable 
j>oint,  its  locus  must  needs  form  a  ])art  of  every  ^-curve  through 
J',  and  these  (b-vnrves  accordingly  must  l)e  degenerate.  By  parity 
of  argument  every  c^-curve  must  consist  of  (y>  —  1)  distinct  j  arts 
where  jf  is  the  common  deficiencv  of  curve  Jf,  and  each  ]xirt  must 
intersect  eveiy  curve  II  in  only  two  ]>oints,  a  conjugate  pair,  out- 
side the  multi})Ie  base  ])oints  of  the  system  [If).  l'"or  an  example 
of  this,  let  the  system  (//)  consist  of  all  cui'ves  of  order  //  having 
in  a  fixed  point  0  a  niulti])le  ])oint  of  ordei'  //  —  '2.  Any  ci-curve 
nuist  have  in  ''>  an  (//  —  .'>)-fold  point,  and  is  itself  of  ord(r  //  —  •'), 
therefore  it  Nvill  consist  of  //  —  .'5  I'ight  lines  thnuigii  ' /.  i\very 
coiisiituent  right  line  has  with  anv  curve  J I  n  —  '1  intersections 
in  O.  and  two  outside  that  })oint ;  the  latter  two  are  conjugate  j)oints 
on  tlie  curve,  ^vhieh  is  conscijuently  hyperelliptic.  Another  ex- 
ample, with  the  <;6's  com])ouuded  of  c()nics,  is  the  system  (jf  curves 
of  oi'der  'Int  -\-  .'>  with  four  fixed  nndti])le  points  of  order  /;*  -j-  1. 
'^i'lie  fact  that  for  these  ]>lane  svstems  the  ])oints  conjugate  to  a 
given  point  (ill  out  a  definite  locus  is  the  thing  to  whi<'h  we  shall 
wish  to  recur. 

In  s])ace  of  three  dimen.-ions,  let  a  surface  /•'  have  all  it<  plane 
secti(»ns  liyj)erelliptic  curves  (''')  of  deficiencv  ji.  ('an  these  lie 
re|)resente(l  by  a  svstem  of  curves  all  in  one  plane?  I-  the  sur- 
face /•"  rational,  /.  c,  tran.--fbi-mable  into  a  plane,  ])oint-tor-point, 
rationallv  V  This  (piestion  again  may  be  approache(l  bv  the  aid 
of  conjugate   ])airs  of  points.      W  e   should   expect   of  course   that 


14  THE    BOSTON    COLLOQUIl'M. 

aiialotriics  of  t/j-cui'vcs  would  he,  in  sj)a('e,  (^-sui'faces,  and  tliat 
those  that  ])ass  throui2;h  a  j)oint  J*  on  any  curve  of  tlie  system 
would  contain  its  conjuixate  O;  and  further,  that  all  ])oints  Q 
conjuuate  to  /*  would  lie  on  some  determinate  curve  of  the  sur- 
face. Tliis  last  suj)])osition  can  he  established  l)y  i-cdncfio  ad 
(ih.snr(hnn. 

The  surface  /•'  and  its  jdane  sections  {(.')  are  algebraic.  Kacli 
curve  '' '  containiuif;  a  selected  point  J*  has,  by  hyj)othesis,  one  ])ar- 
ticular  jtoint  Q  coniu<;ate  to  J\  Therefore  the  C/^  on  the  oo' 
curves  throuuh  J*  must  suit  one  of  tlu;  following  three  descriptions. 

('/)  They  may  be  finite  in  number,  Q^,  (J.„  ■  •  •  Q^.  Hut  then 
every  plane  section  of  the  svu'face  through  7^  would  need  to  con- 
tain   the  line  PQ^,  or  PQ.„  •  ■  • ,  or  PQ,.      Tiiis  is  absurd. 

(//)  The  Q^a  may  be  simply  infinite,  oo'  in  number,  filling  one 
or  more  algebraic  curves  on  the  surface,  or  lastly  — 

{(')  'V\\v  (p's  may  fill  all  the  oc'  ])oints  of  the  surface  /*'.  We 
shall  reject  this  after  showing  that  in  this  case  the  surfiice  must 
be  rational  ;  /'.  c.,  rationally  and  reversibly  transformable  into  a 
])lane,  M'hereas  on  the  contrary,  in  a  ])lane  or  any  other  rational 
surface  the  (^'s  must  be  only  a  sim])l(!  inlinity,  oo'. 

Sup]>ose,  therefore,  that  every  point  Q  of  the  sui'face  is  conju- 
gate to  a  given  /'  ujion  some  one  or  more  (curves  of  the  system. 
It  camiot  be  so  upon  all  ])lane  sections  through  the  secant  PQ,  lor 
then  mu<t  every  j)oint  of  any  ])lane  section  be  conjugate  to  P  on 
that  section,  contrary  to  the  hy|)erelli|)tic  hy])othesis.  Through 
everv  rav  l'(^  there  lie  then  a  finite  number  /■  of  planes  in  which 
/*  and  (^f  are  conjugate.  Anv  one  of  these  determines  all  the  others, 
fbi-  P  and  the  plane  thi'ougli  P  fix  ^,>,  and  the  rest  follow.  Now 
such  a  grouping  of  the  planes  through  /'  into  sets  of /•  plaiu's,  each 
>et  beinu  determined  bv  anv  one  of  its  planes,  is  called  an  invohi- 
rioii.      ( 'a-lcliiiio\-o  ■     proves   the    reinarkal)le   theorem,   ///'//   cvitii 

i lira/ iit/iiii    1,1     llii    iildiiis     (iliniil  a   jxiiiif     in   spthu    iif     f/n-(C  (I I iiH  iisio/i.'^ 

is  fdlhniiil  ;  i.  I.  its  groups  can  be  correlated  imivocallv  and 
n'\-ersibl\-    t'l    the    points    of    a    plane,    each     group    to    one    jtoint. 

Mmi,.   .1,,.,,/,  „.  V..1.    11  1  l-'.M  i,  M..   1 -J.".- 1 :>•".. 


i,in"i;ah  s^■s■l•l:^Is  of  (  ri!Vi:s.  15 

Tliu>  tlifouofli  the  iiu'olution  cvcrv  jxiint  O  of  the  surface  can  be 
related  t(i  -nme  one  j)oint  of  an  auxiliai'v  ])lane,  and  rice  rfr.^a. 
But  if  the  <uri'aee  /•'  he  translbrnied  alii'eUraieally  and  nnivoeally 
into  a  ]i!aiu',  then  its  ])lane  section^  will  l)e  t i-ansfornied  into  a 
liiK-ai'  svsteni  of  hvj)erelli])ti('  curve-  in  that  plane,  conjugate 
])oints  o;i)inu'  into  conjugate  |)(iint>  :  whereas  we  have  seen  that  in 
a  {)lane  the  conjuLiates  (^  of  P  do  not  fdl  the  whole  plane,  hut 
onlv  an  x  '  locus.  Suj)position  (r)  is  thus  dismissed,  and  {h)  alone 
is  tenahle. 

We  have  then  as  a  starting'  ])oint  this  fact,  that  for  a  trencric 
))oint  P  of  the  surface  there  is  a  detinite  curve  />  containing  all  its 
conjuuat"'-  O  tin  the  curves  of  the  system  {(')  ;  and  this  curve  y^ 
can  meet  eacli  curve  (each  j)lane)  only  once  outside  the  point  P 
itself.  If  then  j)  is  of  order  ."?,  it  must  ha\'e  in  P  an  {■•<  —  l)-fold 
])oint.  It  must  also  he  a  plane  cui've  ;  for  a  ])lane  can  i)e  passed 
through  /'  and  two  arbitrary  points  of  y>,  and  will  contain 
.V  —  1  —  1  ^  1  =  N  +  1  ])oints  of  a  curve  of  order  .•<,  hence  must 
contain  the  entire  curve  //.  Tliis  rurve  p  r<ni  he  ■•^/loirii.  to  he  either 
(I  .<frai;//ii  hui'  or  a  ronic. 

II"  y>  i-  not  a  line  or  a  ciMHC,  its  order  ^'  must  be  at  least  o, 
whence  it  mr.-t  have  in  P  [\  double  ])oint(2^.v —  l)ornudti- 
]>licity  of  hiirher  order.  .Vs  [>  is  a  ])lane  curve,  this  means  that 
its  plane  i<  tangent  in  P  to  the  surface  /'';  and  so  that  every  line 
JoininLi'  /'  to  a  conju<rat(^  (^>  is  a  tangent  in  /'  to  the  surface,  and 
l)y  symmetry  of  the  relation  between  P  and  ^^,  tangent  also  to  /•' 
in  O.  '^hi^  is  not  ])ossible  unless  either  the  curve  p  is  a  curve  of 
j)lan(^  Contact  (so  that  P  woidd  be  an  excej)tional  ])oint  of  7*'),  or 
else  the  cui've  y/ c()nsists  wholly  of  sti'aight  lines  through  /*.  This 
altei'uative  is  e(jually  im])ossible,  as  no  ruled  surface  has  through 
every  point  thi-ee  or  moiv  generators.  Therefore  the  hypothesis 
.s'£?.';  leads  to  al)surdity  ;  and  we  have  to  examine  tlie  two  possi- 
ble ca~e~  :   n  =  1  and  ■>■  =  2. 

.V  =  1.  If  j,  is  a  straight  line,  it  does  not  contain  P,  since 
N  —  1  =  ".  To  P  is  associated  one  generator  j)  of  the  rided  sur- 
face /•',  and   conversely,  to  every  point  Q  of  p  must  be  associated 


1<)  THE    1K)S'1X)X    COLLOQUIX^M. 

the  u'cncrator  '/  j);tssiiio;  tlii'outi'h  P.  F  is  f/irn  a  rulrd  sni'Uicr,  of 
liy|)ci-('lli])ti<'  section,  with  its  generators  arrang-ed  in  pairs  cutting 
conjugate  ]')()ints  in  every  plane. 

N  =  '1.  If" />  is  a  conic,  three  cases  can  be  (listinguishe(l.  First, 
to  every  j)oint  (}  on  p  may  belong  a  conic  y  containing  P  but 
different  fi'oni  />,  and  these  may  be  in  themselves  complete  ])lane 
sections  of  the  surface,  If  this  were  so,  the  surface  would  be  a 
(juadric.  lint  the  conies  may  not  be  complete  ])lane  sections  of 
the  surface,  and  this  possibility  it  is  convenient  to  divide  into  two 
])arts,  as  f  )llows  :  Secondly,  the  ^'.s-  may  be  conies  distinct  from 
the  j/s.  The  surface  F  will  contain  in  this  second  case  a  doiih/i/ 
intiiiHc  s)/.sfnii  of  (1  ('(■() ill j)()s(ihlc  or  rcdudh/c  'plane  sections.  Or 
thirdly  (the  only  case  not  trivial),  the  conic  7,  while  its  corre- 
sponding j)oint  (J  describes  the  conic  j>,  may  continually  coincide 
with  j).  There  is  then  only  a  simple  infinity  of  conies  (/>)  upon 
the  surface.  Vo  show  that  this  system  is  a  rational  sheaf,  con- 
sider its  section  by  an  arbitrary  plane  :  on  tlu'  hyperelliptic  section 
curve  each  conic  p  cuts  two  conjugate  ])oints  i^  and  ^^,  and  either 
Z-"  or  (/  determines  j>  completelv,  hence  the  system  ( p)  is  in  cme- 
to-one  relation  with  the  series  of  ])airs  of  conjugate  points  U])on  a 
hvj)erelliptic  ])lane  curve  —  a  linear  series,  and  must  therefore  be 
rational.*  Now  tliese  tliree  altei'uatives  lead  to  a  single  conclu- 
sion, tiu'ough  the  application  of  well  known  theorems. 

Fii'st  if  the  sui'face  /•' were  (piadric,  it  would  l)e  rational  ;  but 
then  it  wiiuhl  be  discussed  as  a  surface  with  all  its  plane  sections 
I'ational.  i'       I'^or  the  secfuid  case  we  adduce  K roneeker's  theoi'cm  T 

liiilci-tl  tlic~i  |iliiic^  t'iifin  the  ilrVfl(i|i:iliK'  (if  ;i  twi-tcd  culiic  (■iir\i'.  since  no 
Miir  (if  liiriii  1-ciiiiit-,  twice  :  ( ';i>teliiiinvii  sliuws  that  tile  i 111 iiiei  1  iai e  i;enera]i/.atiiiii 
i.l'llii-  reiiiai-K   Imld--  t'.il'  li  \pel~liace. 

'  See  papii'  liy  v..  I'icaid  :  "Siii-  les  -iirt'acev  alL:i'liri(iiie>  iliml  tuiiie^  le<  sec- 
tion- plane-  soul  uiiiciir-ali-."  <'r'ili'.<  .Iniininl,  vnl.  1  0(1  (  1  SS.')  ) :  and  a  cdiTelated 
liaperiil'  K.  II.  MiM.ie:  '  ■  A  li^elii-a  ie  -lui'aco  nf  wh  jeli  cverv  jilane  sect  idn  i<inii- 
eiir-al    in    liie    li-lit    ..t'    ;< -ili  nieii-ii  ,na  1    -vdinet  ry,"    Ann,-.    ./..,/,-.   ../'   Mull,..   \n\.    lo 

(    I--S  :.    .,.     17, 

;  Sei-  till'  li  i-Ii  iiira  1  ni  ite  and  deim  m-t  lal  ii  m  li\  (  'a-.Ielniii  i\(i.  "  Snl  le  --npeliieie 
al  L^'^liciclie  cln'  ainniettniKi  un  si-teina  d' ippianienti'  inlinitn  di  se/.i-ini  jiiane 
ridnltiliili,"    /,',-.•-/■  II.  n-lirnnli.  .lanuai's-,    \<'.A. 


LIN'EAII    SYSTEMS    OF    CUK\i:s.  17 

that  ((  stii'i'dce  hdriix/  a  donhfc  iiifi)iiff/  oj  j>/(iiic  .sci-tioii.-^  that  arc  <lc- 
coinposdhle  ciiitcs  is  cithei-  <i.  St('in('j'\s  (iticrtic  sii/'Jacc  or  n  ruli'd 
Hiii-facc.  —  Steiiicr's  "  Ivoinati  Surface"  is  the  (jiiartic  having  tliree 
douhle  lines  tliron<;-li  a  tripk'  point,  and  is  rational,  since  it  can  he 
projected  U'<nn  its  triple  point  upon  a  plane.  The  third  case  is 
decided  1)V  Xoeflit'i-'s  theorem  *  that  d  siirjdcc  coutninliKj  a  /■afioiKtl 
si/i<(('iii  of  fdtioiuil  curvi's  is  ratioiKi/. 

The  conclusion  can  he  condensed  now  into  tiie  form  :  I'Jrcrt/ 
(ih/rhrair  i^iij-fccc  iv/iosc  j)/<ine  scctionx  arc  Inipcrcllipfic  i-nrrcs  of 
(h'fjch'ncji  J  or  more  ix  cUher  (./)  a  rnled  xnrfticc  or  {./')  <(  r<ffion((/ 
siirfocc,  and  in  (lie  Idftcr  (ilferiidiire  if  coiddinx  d  rdtioridf  -^/tcdj'  of 
coiiirx.  This  latter  ])hrase  obviously  rejects  two  of  the  alterna- 
tives ol'  the  j)recedin<>:  })aragraph,  and  this  is  warranted  hy  the 
rationality  of  the  surface,  the  representative  system  of  plane  curves 
Ijeing  tlierefore  the  criterion.  For  we  recall  that  in  a  linear 
system  of  ])lane  hyperellipties  tlie  j>-curves  and  7-eurves  discussed 
above  are  comjtonent  ])arts  of  the  degenei'ate  (/)-(airves,  and  a  j) 
coincides  with  all  its  7's. 

This  highly  general  theorem  allows  us  to  study  upon  plane  svstems 
the  geometry  ujxui  an  extensive  family  of  surfaces  in  space  and  in 
hyi)erspace  as  well,  since  the  existence  of  a  triply  intinite  linear 
system  of  hyperelliptic  curves  in  a  surface;  is  e(|uivalent  to  the 
hy])othesis  that  we  liave  used  concerning  plane  sections  in  ordi- 
nary space.  And  ibi-  linear  systems  of  the  first  kind  in  a  plane 
reduced  typ(;s  have  been  found  by  Castelnuovo,'?"  from  which  all 
others  are  derived  by  ( "remona  transformations.  It  I'cmains  to 
develop  to  the  same  extent  the  theorv  of  svstems  of  the  second 
kind.  This  would  demand  an  acipiaintance  with  the  work  of 
liertini  on  plane  involutions  of  index  2,  and  of  ( 'lebscli  and  ^Oether 
on  I'ational  double  planes. 

An   extension    in   another  direction   has    been  given   by  ( 'astel- 

t  M.  Ndctiu'i-  :  ■'  I'c'hcr  l''liiclicn,  wclclio  Si'lKUU'cn  nitioiiaU'r  ( 'iir\  (,-ii  hesit/.eii,"' 
M'llh.  Aitii'ih  II,  vol.  ;')  (1S71).  pp.  17.">-1.  Tlie  tlit'orcm  i<  iiiort'  <;\'iU'ral  tlian 
that  licrc  citt'd. 

■■■  "Sullc  supei'tifif  al£(oliric]i<'  Ir  cni  sc/.ioiii  piain'  sciiin  curve   iperfllitticlu'." 
Pdh'riiri  UrmUfon'i,   vol.    1   llS'M)),   pp.  To-HS. 
o 


18  THE    BOSTON    COLLOQUIUM. 

nuovo,*  wlu)  (lisoussod  under  only  one  s])ecializing  restriction  the 
surfaces  wiiose  ])lane  sections  are  of  deficiency  3.  These  are  of 
four  kinds,  so  far  as  numerated,  and  not  all  rational. t  In  lookin*^ 
for  other  ])ossihle  extensions,  it  should  be  remembered  that  tliei'e 
are  other  classes  of  hiuhly  s})ecialized  curves,  differing  from  the 
hyperelliptic  in  the  degree  of  the  singidar  series  of  special  grou])s 
which  occurs  upon  them.  Of  such  (dasses,  individual  curves  have 
received  some  study,  but  linear  systems  little  or  none. 


ClIAl'TEIi   4. 

Jjiiicdj'    Kxarf    DiffvrciitUdx    of   the   FlrM    Kind   on    (in    A((j<hraic 

Siirjdcc. 

^  1.       The  Kxi!<f('n<-e  of  Infer/i-(ds  0)i    (Ih-en    Siir/<(cex. 

When  the  theory  of  integrals  Uj)on  algebraic  curves  was  ex- 
tended to  surfaces,  the  first  step  was  the  discussion  of  double 
integrals.  These  have  been  described  already  (C1ia])ter  ]),  and 
attention  has  been  called  to  two  im])ortant  numbers,  characteristic 
of  a  surface,  to  which  they  give  rise,  the  geometrical  and  the  numer- 
ical deficieucv.  Kvery  surface  above  the  lowest  orders  ])ossesses 
double  integrals  of  the  first  kiud,  everywhere  finite,  unless  its 
siugnlarities  have  become  too  numerous,  '^l^he  iucrease  of  singular 
point-  aud  liues  <'auses  a  diminution  of  the  geometrical  deficieucv, 
y>  .  Double  iutegrals  and  their  classification  were  inti'odueed  by 
Clebseh  and  Xoether  about  1S7<>.  k'ifteen  years  later  a  different 
and  even  more  interesting  extension  of  curve  theorv  to  surfaces 
was  UKule  bv  Pieai'd.t  The  new  iutegi'als  that  he  introduced 
are   simple    integral-^    \vlio~e    path    ot'    integration    is    restricted    to 

'    "Silllf    ^lljirllicic    IllLTfliricllr    If    clli   -c/.idlli   SnlK  i  ell  I'^'C  lii    UCIlcrc  •">. "       Tiilil>() 

All  I.  vol.  -i:,  I  l^'.Hi ,. 

i  It'  t  III-  -iirf:iri'  i-  111'  (ii'ilri-  aliiive  t  Ik-  f"ii,  t'l,  with  jilaiu'  sect  inns  ;ill  nt'  ilcticii-ncy 
;!,  it  i-  |-.itinnal.  Scr  ( ';i>t(lniiii\i  I  ainl  l'!iifii|Uis  "  Supra  alcmic  i]Ui'sti(iiii  feii- 
ilaiiii'iitaii    ii'Ha    troria   ilrllr    ^iipcrticic   aiuflirirlic,"    Aiiniili  ili  Mnliiiiiitirit,  ser. 

:;,    Vnl.    f,    (    l',.(ll    ,.    ,..,,.    Sec.     \-,    ^   It',. 

;  "Mir  1.-^  ini>'L,'i'al(-  ill'  ililii-rcnticllr-  tot  air-  ali,'i'liriijiics  do  ])reinirrr  csiircc," 

./„»,.    ./.     „..-/',.,    -IT.      1.    V..1.     I     [    1>S.')). 


LINEAR   SYSTE:MS   OF    CI'RVES.  19 

lie  in  the  surfac^o,  while  the  inte<^ra]s  arc  furtlicr  required  to  he 
fiiiictious  of  tlieir  limit  points  alone,  not  of  the  })articular  path  of 
integration.  The  number  of  linearly  indt'pcndent,  everywhere 
finite,  integrals  of  this  kind  is  a  new  invariant  characteristic  of 
the  surface  ;  and  it  is  found  that  this  number  is  zero  when  the 
surface  is  non-singular,  but  increases  (according  to  a  law  not  pre- 
cisely known)  with  the  multiplication  of  singularities.  This  is 
the  theory  of  which  ]  propose  now  to  give  a  sketch,  following 
very  closely  Picard's  article  cited  above.''' 
Upon  an  algel)raic  surface 

f  =  f{.r,y,z)  =  () 

a  linear  ditt'erential  expression  in  d.v,  dij,  dr:  can  be  reduced  l)y  the 
use  of  the  relation  : 

By  this  means  any  expression  of  the  form 
AtJ.v  +  lull/  +  Cdz 
may  take  on  either  one  of  the  three  aspects  : 

.,  dz  + dx, 

J    y  J    1/ 

^.,  dx+  ^.,        ■  dlJ, 

Let  tlie  first  be  chosen,  and  abbreviate  it  to 

Q,]x  -  Pdii 

J"-: 

Concerning  this  ex])ression  two  things  are  to  be  noted.  First,  if 
the  surface  be  cut  by  an  arbitrary  ])lane,  then  by  the  adjunction 
of  the  equation  of  that  plane  this  must  become  an  abelian  differen- 
tial of  the  first  kind  upon   the  plane  curve  of  section.      Secondly, 

*  J'^or  details,  see  also  tiie  hook  of    I'ieani   and  Siniart. 


•_!<) 


TIIK    1U)ST().\    C'OLlAMnil'M. 


eitlu'i"  ,/•  oi"  //  could  luive  bcon  taken  as  dependent  variable  instead 
of  :. 

Fn)ni  the  Hrst  observation  it  follows  that  !)oth  P  and  Q  must 
!)('  entire  functions  of  ,/•,  //,  :,  of  order  //(  —  2  when  //;  denotes  the 
order  of  the  sui'face  /"=  0. 

From  the  second,  convertiuii  the  differential  into  its  two  ecjuiv- 
alent  forms  : 

—  (I.r -\-  Id',  —  .,  (11/  —  (/(iz 

./-  _  J. 

./■;  ~  /■: 

we  Hud  further  that  the  fractional  form 

Q.f:  +  ir.. 

J'r. 

must  reduce  to  an  inte_u-ral  form  upon  tiie  surface,  /.  c,  by  virtue 
of  the  equation  of  the  surface.  TiCt  —  li  denote  this  integral 
form,  and  A'  a  suitable  j)olynomial  of  order  m  —  '■>,  so  that  we 
shall  have  identically  : 

(^)  /'■/:+  Qj':,  +  /'!/■;  +  a-- /•=(». 

'Phis  gives   us   fir  ('(juivalent   dillcrential   expi'cssions  on  the  sur- 

i'acc  : 

y    ,/.,.  _  /> .  ,/y       />,!,,  _  O  ■  ,1-.        I\l:.  _  /.',/,,■ 

'Hk'I'c  is  yet  to  ai)p]\-  the  c()iidition  for  an  exact  dillcrential,  in 
oi'(ler  that  the  integral  between  anv  limiting  |)oints  ina\'  be  a  func- 
tion oftlio^c  limits  independent  of  the  path  of  integration.  That 
condition  in  one  foi'in  will  be,  upon  the  surl'ace  : 

I'ei'foi'niing  tlic-f  ditl'ri'cnl  iat  i(  in-  b\-  the  aid  of  (  1  ),  and  multiply- 
ing b\-  (  /"    )-'  \\r  Ii;i\-c  for  /'  =  1 1  : 

CI'     ro     ci: 


ijxi'.AK  s^■s■|■I•:Ms  of  ('ri;vi:s. 


21 


Tills  must  liold  over  tlio  surface   /"=  0,  liciiee  usiiiLT  (-)  and  cuin- 
plotiiiu'  the  al<rel)i'ai('  identity  hy  a  term  In  j(.r,  //,  :)  nvc  find  : 


(•') 


wlicre  I J  denotes  some  intei^ral  functi(»n  of  .r,  //,  :,  of  order  2///  —  (). 
IvKiianding;  in.  part  the  third  term,  we  distinguish  terms  wliich  on 
their  face  must  contain  a  factor _/'(.r,  //,  :)  : 


\  ex         CI/         C-:  J  V  -     ,/  /   7 


=  0. 


Since  the  first  grouj)  of  foui'  terms  are  integral,  and  of  order  lower 
than  ///,  they  cannot  contain  tlie  factor_/'(.r,  _y,  :)  otherwise  than  by 
vanishing  identically.  Tims  we  must  have  for  all  values  of  .7',  //,  z 
the  identitv 


(•5) 


cP      c(.)      CI! 

c.r         CI/         c~ 


Insert  again  the  equivalent  of  .\'_/"from  (2)  : 

(7)    p  ■/:  +  ^  ■./•;  +  //  ■./•;  ^m  h,  ■-)  ■  (^'J^.  +  ^^  +  '^' 

The  form  of  this  identity  invites  us  to  write  /  homogeneouslv 
in  (.'■,  //,  :,  /),  —  and  of  course  the  other  functions  also,  and  to 
em[)loy  luiler's  identity 

ii{f=.rj'  +  i/fi  +  v/V  +  //;, 

so  that  e(|uation  (7)  becomes  : 
('"P  +  ■<'X)J'[  -f  (niQ  +  i/\)Jl  ^  [ml!  +  .:.V)/':  +  tX-j]  =  (I. 
In  this  it  will  apjjcar  more  simjtle  to  write 

,„ r  +  j'X  =  fO^,    ui  (^  +  II y  =  (0.,    ill /;  +  :.v  =  te.^, 

(S) 

x=e^. 

T(»  show  that  ^|,  ^.„  ^^j  are  integral,  recall  that 

Q,J,r  _  /',/_;/ 


ii2  THE  BOSTON   coLLO(H'iu:\r. 

is  a  total  differential  nowliere  infiiiite,  and  that  in  tlie  plane  /=  0 
it  is  an  abelian  integral  of  the  first  kind,  and  so  mu.st  have  the 

form 

ydx  —  xdij 
4>{x,y,z)-'-       ^.,       ■, 

where  cf)  is  of  order  //(  —  ■).      For  this  reason  we  must   have  iden- 
tical Iv  : 

where  (f)  is  homogeneous  of  order  m  —  .")  in  .r,  ij,  z,  and  ch^,  cp.,,  0.^ 
are  of  order  id  —  3  in  .r,  i/,  z,  and  /.      Therefore 

/dp      c'Q      CR 

c'.r  fi/  cz 

or 

=  —  ///^  +  f(f)^. 

T}ic.'<e  exjircxti'ioiu^  (jlvc  fo/'  f/ic  ^'n  fhc  i iiter/nii  fitruis : 

e^=  ^  =  >ii4>^  +  ■'■<^, 

6.,  ■=  ,n(f).,  +  i/(f)^,      d„  =  111(f).,  +  .:•(/),,      ^^  =  —  iit(f)  +  ^4>,- 

l']ff('cting   the   substitutions   (S)    in    conditions  ((i)   and    ( 7  K  and 
using  Kuler's  rel.'itiun  for  A': 

c.v       c\      cx  r\ 

C.I'  ii/  cz  ci 

we  have  the  two  relations  which  the  ^'s  must  satisfy  : 

CO,     re,     CO..     co, 


c/  (  /  (  /  (  / 

(!<•)  ^,  :    -^0.,:    ^0..'  -f  ^,  :    ^  <•. 

'riie-c  conditions  ai'e  now  ^N'mmct  rical  in    the  tour  JKniioi^ineous 
varinMo.  :ni<l    l)v    the  aid    of  foiu'   jiai'ametcrs   <-^,  <■.,,  <■,,  .-^  we  can 


IJXKAR    SVSTK.MS    OF    (TKYKS.  23 

Ix-iiiij:  tln'  int('i2;ral  (•'))  into  tlie  symnu'trical   form  used  by  Jiorry.* 


(11)  u  =  j 


e.. 

0., 

^, 

1/ 

/ 

i'f 

<h 

<n 

dx 


Tliis  integral  is  suhjeet  of  course  to  the  further  restriction  of  re- 
maining finite  in  all  singular  points  or  lines  of  the  surface.  Prima 
f(tcic,  the  ])resence  of  singular  points  or  lines  seems  a  restriction 
u))on  the  number  of  linearly  independent  sets  of  polynomials  6. 
But  the  fact  is,  that  if  the  surface  have  no  singularities,  it  can 
have  no  such  integrals.  For  by  tlie  identity  (10)  the  (//;  —  1)^ 
intersections  of  three  first  pohir  surfaces  : 

f'.,  =  ^\  /:  =  '»,  ./;  =  <^ 

must  fall,  eithei-  on  the  surface/'^  =  0,  thus  indicating  a  multiple 
point  of  the  fundamental  surface  ;  or  if  not  one  fall  on  this  fourth 
])ohir,  then  all  nnist  needs  lie  on  the  surface  6 ^  =  0.  This  last, 
however,  is  of  order  m  —  •),  and  its  ccpiation  cannot  be  a  linear 
coml)ination  of  three  e(juatious  of  order  //(  —  1.  And  the  conclu- 
sion cannot  l)e  escaped  by  suj)posing  the  polars  :  /'  =  0,/'  =  0, 
Jl  =  ()  to  have  a  curve  in  common,  since  then  that  curve  must 
pierce  in  a  niuuber  of  points  (or  else  lie  wholly  in)  the  fundamen- 
tal surface,  ireuce  the  surface  _/' =  0  must  have  at  least  multi})le 
points,  as  was  to  be  proven. 

Whether  a  surface  whose  equation  is  given  does  or  does  not 
possess  linear  differentials  of  the  first  kind,  and  how  many  linearly 
iiuh^lU'udent, — this  can  be  determined  by  first  finding  the  luul- 
tiple  curves  and  jxjints,  and  then  counting  the  conditions  imposed 
by  them  and  by  the  identities  (G)  and  (7),  or  (9)  and  (10). 

liCiiurr/:.  The  inunber  and  nature  of  singularities  that  a  sur- 
face; of  given  order  must   j)ossess  in   space  oi"  three  dimensions  in 

-  Tnnis.  <}nnhriily>-  I'hU.  S„r.,  vol.  18  (1900),  p.  :;;5:5-l. 


'24  riii:  ijosrox  coi.ixx^uiu^tr. 

ordci'ln  admit  (iiic,t\v(),  •■  •,  iiKlepciuU'iit  exact  differentials  of  this 
first  kiiul  iiii<;lit  ])r()V(' an  accessible  and  jirofitable  snbjeet  for  fnr- 
tlicr  iiHjiiiry.  'i\i  extend  this  incjuiry  to  suriaces  in  hyperspaee 
would  i'e(|uii'e  a  systematic  preliminary  study  of  curves  and  sur- 
face- in  such  a  space  not  yet  completed. 

§  -.     Tlw  /■^.ri.s!t'ii('c  of  .'^i/rj(ires  of  (rireii   ( '/Kirdcfcr,  In  pdiiirukir 
lljlixrcllipiic  Siirfdcc-^. 

If  a   surl'ace   in    threefold   si)ace,/=(>,   possesses    exactlv   two 
lineai'lv  independent  exact  differentials  of  the  first  kind 

Qfi.r-I'rlf/  Qd.r-l\<h, 

=  (hi      and       .,  =  (Iv, 

J.  J 

then  everv  alu'chraic  curve  lyinL!;  npon  it  has  the  same  two  inde- 
pendent ahelian  diflerentials  of  the  first  kind,  and  hence  these  in- 
tegrals have  four  inde})endent  sets  of  periods.  It  can  be  proven 
easily  that  the  geometrical  deficiency  of  the  surface  is 

.iind  that  the  exjji'cssion 


// 


J":  J": 

IS    the   double    inteu'ral    of  the   first   kind,  finite   for  all    boun<larv 
curve-  on  the  surface. 

( 'ou\<'rscl\-  PicAlM)  shows,  (/.  c.)  that  if  ,r,  //,  :  are  o-jvcn  as 
fourfold  j)criodi<'  lunctions  ol' two  indej)cndent  vai'iables,  the  locus 
of  a  point  (.'■,  //,  : )  is  a  sni'face  of  this  sort.  l'^)r  a  simplest  illus- 
tiatinii  let  the  finictious  reduce  to  elliptic,  and  in  the  W'eierst rass 
notation  set 

-<■  =  ]>(  I'),     .'/=}>('•),     '-=  l>'{")  +  i>'(s)- 
'Ihi-  i;i\'es  an  ((piation  between  .r,  //,  and  :: 

'  "^   I    ^■'"'  —  .'/■_.■'■  —  .'/■.  +   I     ^!f'  —  !/■>'!  —  !/:■: 
or  toi-  bre\it  \- 


LixKAK  sYS'ri::Ms  OF  (  ria]:s.  ti") 

A('('<)i'(linolv  tlie  surface  is  of  tlie  sixth  order  : 

4/.'(.r)  •  R(j/)  -  [,"  -  //(>■)  -  h\,/)y=j\.r,  1/,  ,:)  =  0. 

The  t\vo   integrahs  on   the   surface,  //  and  r,  are   rejtresented   as 
follows  : 

"~  J     \    JUx)  ~  J 


f 


(Ja 


,    7.'(.r) 
r  ^    \  =1  ■  ..  ■  (III. 

As  to  double  integrals,  the  one  of  the  fii'st  kind  belonging  to 
tiie  surface  degenerates  into  /  dii-dr^  which  is  evidently  finite. 
The  double  lines  of  the  sextic  surface  may  be  perceived  imme- 
diately, one  of  them  being  obviously  the  straight  line  .i'  =  //, 
,-.  =  (»;  another  a  conic  in  the  (•'',.'/)  plaiu^ ;  and  three  lines  at 
infinity. 

One  linear  diiferential  of  the  first  kind  can  exist  on  a  surface 
of  order  as  low  as  the  fourth.  There  are  iive  tyj)es  of  such  quartic 
surfaces,  found  by  Poincare,"^'  ]]erryt  and  de  ]*^-anchis  ;  the  five 
ty})es  are  ])rojectivelv  distinct,  that  is,  collineations  cannot  trans- 
form one  into  another;  but  I>erry  has  found  that  under  birational 
transformations  all  five  are  e([uivalent  to  a  cid)ic  cone  devoid  of 
doid)le  line. 

Of  these  five  types,  pei'haps  the  easiest  of  derivation  is  the  fol- 
lo^ving.  The  <|uantities  6,  being  of  order  //(  —  •">,  are  linear.  Let 
their  ])lanes  coincide  with  those  of  the  tetrahedron  of  refei'cnce,  viz.: 

thus  satisfying  the  condition  (!•).      It  remains  to  satisfy  (1(V), 


'7  ^      ^.7  ^7  cj 


a 


*('nuii>l>'^  J!e,i,lHs,  vol.  W  (Dec.  2'.),  18S-f). 

t  Ihiih'tit,  Sejit.  ■_',   lS',t',).      See  also  liis  papers,  ciU'il  above,  in  the  Trini.^.  ('citt- 
hrl'l-jr  ri.il.  Sur. 


'2()  THK    BOSTON    COLLOQUIUM. 

Since  also  by  iMiler's  identity  for  homogeneous  functions 

cf         Cf         Of        of 
c.r      ''  cjj         cz         ft         •  ' 
it  follows  that 

^y  y         y       /Y       ,^ 

c.r  cij  cz         ct 

Hence  the  form  is  homogeneous  of  order  2  in  .r  and  _?/,  also  in  :: 
and  /.      Svnibolically 

/=  {<t/-  +  •l<i,,r,i  +  ajr){l>^  +  2M  +  b.f), 

each  product  <i hj,  denoting  an  arbitrary  real  (juantity.  This  is  a 
familiar  ruled  (juartic  surface  with  two  double  lines  (.r  =  ?/  =0 
and  z  ^=  t  =  0).  It  is  generated  by  taking  for  directrices  these  two 
double  lines  and  any  plane  quartic  which  has  nodes  upon  the  two 
lines. 

This  suggests  the  inter])retation  of  conditions  (0)  and  (10)  by  a 
complex  of  lines.      The  connex 

gives  I'ise  to  a  complex  when  every  })oint  (.r)  is  joined  to  its  cor- 
responding ])oint,  and  condition  (H')  is  the  re(piirement  that  the 
comj)lex  line  oi'iginating  in  a  ])oint  (.r)  of  the  surface  _/'=  0,  shall 
be  tangent  to  that  surface.  Si)eaking  of  the  line  joining  a  j)oint 
(.r)  to  its  corresponding  point  in  the  connex  :  ^  =  0  as  a  ii'(ij(cfor>/ 
of  that  connex,  we  say  :  .1  Kin-face  J  =  0  of  or(l(r  ni  iri/f  j/osscsk  a 
/iiirar  cruel  (liffci'ciitidl  of  f/ir  flri^f  hind  if  <i  conijili.r  (/;/  —  ."i,  1) 
c.risfs  such  that  t/ir  IrdJccfnrK's  of  (ill  points  (hi  llic  xii rfdci  >' re  Ui iKjcnt 
In   l/ii    siirfiicc. 

1!(  iiiiirl:.  W'licii  (uw  lineal"  exact  differential  exists  on  the  sur- 
face, and  onl\'  onc.it  is  invariantively  related  to  the  sui'fac<'  uudei" 
a  much  hirgrr  group  than  that  (»f  the  coJlincation<  and  a  fnrliari 
inidcr  ihc  latter  uroiip.  Instead  of  .-.eck'iiig  tlic  integral  when  the 
-iii'tai-e  i-  gi\'en.  and  finding  it  as  an  ii'rational  covariant  of  the 
-.iirl'aee,  mH''   niii^ht    attempt    to  dcti/rmiue  the  surface  a-  a  rational 


IJNKAR    SVSTi::\IS    OF    ClliVKS.  '2i 

covariivnt  of  the  forms  6^,  6.,,  6.^,  6*,,  occiirrinir  in  the  intoural.  ]>ut 
the  surface  may  be  not  determinate.  (In  the  above  exam])le  it 
had  still  8  free  parameters.)  ^Vlso  the  ^'s  depend  on  the  choice 
of  planes  of  reference.  Hence  more  ])recisely  one  should  seek  to 
determine  the  mixed  form  f(.r,  n)  (connex)  covariant  with  the 
connex 

d  =  u^e^  +  uA  +-  '^.A  +  ".^-. 

such  that  every  set  of  values  (»)  makes  the  surface  /(,r,  */)  =  0  a 
surface  ])ossessino;  the  integral  of  the  first  kind  represented  in  (11). 
In  other  words  the  connex  6  is  to  satisfy  the  relation 

dW         d-0         c-e         c-e 

c'«|C'.rj       cn„cx,^       cii.^c.r.^       cu^c.r^ 

w'hile  the  covariant /(.r,  ?(),  or  f,  is  to  be  of  order  in  the  (.r)  higher 
by  3  than  6,  and  shall  satisfy  also  identically  the  equation  : 

DO    cf        cd     cf       dd     cY       dO     cY' 

.     .•    +  .     .•    +  .     .•  ■  +  .     .•'  ^  0. 

Of  course  the  chief  interest  in  this  problem  would  be  found  in 
the  lower  orders,  4,  5,  (].  It  might  be  ])ossible  to  solve  a  similar 
prolilem  of  the  theory  of  forms  when  the  surface  is  to  have  two  or 
more  independent  integrals  of  the  first  kind. 

To  return  to  surfaces  with  two  indej)endent  exact  differentials 
of  the  first  kind,  we  note  two  theorems  of  Picard.  T/ic  e.vi.'<tenee 
of  two  sHcli  (litf'ereidioU  /.v  haposf^ihlc  iijK)it  din/  i^io-face  of  order 
hi^V).  If  a  .sio'fice  have  tiro  .such  diffeirritla/x,  its  phnie  sectio));^ 
(ire  curves  oj'  <feficien<\i/  at  least  p  =  2,  (did  ifx  geometriral  (teficieitri/ 

'•-^  1',^  f. 

Picard  establishes  directly  the  existence  of  a  class  of  surfaces 
with  two  differentials,  in  brief  as  follows  :  Let  the  Cartesian 
coch'dinates  of  a  point  be  given  as  uniform  functions,  (piadruplv 
[)eriodic,  of  two  independent  variables.  Let  the  relation  l)e  such 
that  to  every  point  (.r,  //,  -)  of  the  surface  there  corresjionds  (Uie 
and   only  one  pair  of    values  of    the  two  indcj)endent    variables 


■2S 


TJIK    1K)ST()X    COLLOQlir^r. 


II,  r.      Tlicii  the  surface  has  exactly  two  linear  difFerentials  of  tiie 
tii-.-t  kind.' 

i''(ir  if  the  surface  ecjuatiou 

./■(•'■, .'/,  ^)  =  0 

is  satistieil  identically  hy  three  uniform  functions 

and   the   functions    J\,  F,,  F.,  have  four  simultaneous  systems   of 
j)eriod-.  then  since 

cF,  dF, 

<J.r  =    ,    '  ,1,1  +    ^   '  ,lr 
CH  cr 


rF.,  cF., 

(hi  =    ,    "  (hi   -f-     ^   '  (h 
ill  cr 


these  pai'tial  derivatives 


Cii    '  '    cr 


niu-t  he  likewise  (juadi'uply  j)erio(lic  uniform  iunctions  of  ",  /•,  and 

therefore  rational    I'unctions  of  ./■,  y,  :.      Accordingly  tlie  solutions 

of  the-^e  two  ('(juations 

fJl,  =   Q^,J.r  -   /\<l;/ 

rlr  =    (Jjl.r  -    I\<hi 

ai'c  differentials  of  tlie  lirst  kind  ii])oii  the  surface,  and  independ- 
ent hv  hyi)ntlie-is.  Ihit  aiiv  third  diffei'ent ial  of  this  kind  on  the 
surfa<'e  i^  neee~-arily  a  linear  finietion  of  these  two,  with  constant 
ci  lenieieiit-.       If  it  he  denoted  1)\'  '//'•; 

,ln-  =    Q,,l.r  -    /'//,/ 

=    (   (J    ^     <hi  _   /',  J    ,lu\    I        (),    ,     ,lr   -    /',  ,•     <h 

\    ^  ■  (11  ■en        J         \    '  ■■  (  /•  ■■  (  u 

=    (f>{.r,    V,    \]ihi  -f-    \//-(.''.   //,    •-)''''•, 
I. ^,,,,,,11.,  ...I-.    !.   v.. I.    1    I  ISs:,  ,. 


ijXKAR  svsti:ms  of  crinEs.  29 

then  uri  tlic  functions  cf)  and  -v//-  an,'  assumed  to  remain  finite 
throuf>'hout  t!i(,'  surface  /'=  i),  and  are  seen  l)y  the  f'oreixoinij:  to  he 
rational  in  .r,  i/,  z,  they  can  be  notliiui^  but  constants,  as  was  to  be 
])roven.  'J'he  d<juble  intciLrral  of  first  kind  on  the  surface  is 
/  jiludc;  the  proof  that  it  is  unique  is  closely  similar  to  the 
above.  Functions  of  the  j)roperties  re(juired  for  /•',,  F„  l-\, 
are  readily  expressed  by  quotients  of  theta-funetinii-  of  two 
variables. 

Suriiices  of  this  sort  are  called  by  Picard  and  Humbert  lii/jxr- 
('//ijAic  ^mrfarcs.  They  are  to  be  distiniruishcd  carefully  from  sur- 
faces whose  j)l(iiic  sccfioiis  are  hv[)erelliptic,  or  which  have  a  linear 
net  of  hy[)erelli])tic  curves  ui)on  them,  for  those  we  have  seen  to 
be  rational  (y>  =  t'j;  while  these,  possessing  one  double  intci^ral 
everywhere  finite,  have  y>,  =  1. 

IIy[)ei"elliptic  surfaces  of  order  lower  than  the  sixth  do  not 
exist,  as  was  said.  This  evokes  recollections  of  Kummer's  sui'- 
face  of  the  fourth  oi'der  ;  but  that,  as  Picard  shows,  is  not  of  this 
cla-^s,  because  it  has  firo  srtx  of  values  (",  r)  for  every  ])()int  (.'■,  y,  : ). 
llniiihcrt  lias  discussed  hyperelliptic  surfaces  //(  e.rfensh/''  in  par- 
ticular those  of  sixth  order.  lie  extends  this  mode  of  establishin*^ 
their  existence  by  theta-formnhe,  so  as  to  emplov  the  next  higher 
class  of  thetas,  those  in  three  independent  variables.  Jji  this  way  he 
reaches  sui'faces  containing  three  linearly  indej)endent  exact  linear 
dififerentials  of  the  first  kind  and  ])roves  that  their  order  must  be 
highei'  than  six.  An  example  is  given  of  the  eighth  ordei',  l)ut  the 
order  seven  is  left  in  doubt.  Of  such  representation  of  these  sur- 
faces, the  chief  advantage  is  that  ev(>rv  algebraic  curve  lying  in  the 
surface  is  given  by  the  \'ani.--liing  of  some  theta  function,  so  that  bv 
the  use  of  theorems  more  or  less  familiar  in  the  theorv  of  thetas, 
one  o!)tains  an  exhaustive   treatment  of  geoinetrv  upon   a  sui'face. 

It  is  apparent  that  this  line  of  investigation  o])ens  a  prospect 
of  a  classification  of  surfaces  based  on  properties  nnich  moi'c  ltcu- 
eral  than  those  merely  projective.  As  was  indicated  in  a  remark 
upon  ((uai'tics,  this  calls   for  the  projective  study  (for  the  sake  of 

-^ /.;,"-■///,,  s.T.    !,   vol.  .',  (hSS'.t),  v(.il.  ',»  (ISiiM.,  and  ser.  ■',,  v,,l.  2  (IS'jf,  ,. 


oO  THK    ]5()ST()X    COLI.OQUIU^I. 

models)  of  .surfaces  which  become  interesting  under  this  more 
searchiiiii;  light.  And  the  special  classes  —  as  those  related  to 
point-jniirs  on  one  curve,  on  two  curves,  those  in  which  the 
pn-iods  of  the  arguments  fall  into  some  integral  relation,  etc. — 
those  otfer  a  field  most  invitin<x  and  likclv  to  vield  rich  fullness 
of  even  the  simpler  geometric  forms. 


FORMS    OF    XOX-FFCLIDFAX    SPACF. 

Bv  FKKDKRK'K    S.  WOODS. 

J>y  a  iioii-euclidean  o^eonictrv  we  shall  mean  any  system  of 
geometry  which,  while  differing  in  essential  ])articnlars  from  that 
of  Fnelid.  is  nevertheless  in  accord  with  the  facts  of  experience 
within  the  limits  of  the  errors  of  observation.  The  space  in  which 
such  a  geiunetrv  is  valid  is  a  non-enclidean  space.  It  is  clear  that 
the  test  of  ex[)erience  can  l)e  aj)plied  only  within  a  restricted  por- 
tion of  space,  so  that  non-enclidean  spaces,  while  having  essen- 
tially the  same  properties  in  such  a  restricted  region,  may  differ 
widely  when  considered  in  their  entirety.  It  is  the  purpose  of  the 
present  lectures  to  present  es])ecially  those  non-euclidean  spaces, 
investigated  by  Cliftbrd,  Klein  and  Killing,  which  have  been 
named  by  the  last  autlior  the  ('/iJor(1-K/('in\^c/ie  Ji(i}iii(J'o/'i/u'ii.'--- 

For  the  sake  of  clearness  it  is  necessary  to  begin  with  the 
geonietrv  of  a  restricted  portion  of  space.  Here  the  author  has 
followed  the  development  of  his  own  article  on  "  S])ace  of  Con- 
stant ( 'ur\-ature,"t  to  which  the  reader  is  referred  for  references 
to  the  literature  and  for  fuller  handling  of  some  of  the  subject 
matter  of  the  first  five  ])aragraphs  of  these  lectures. 

The  j)()int  of  view  adopted  is  that  objective  space  presents  cer- 
tain phenomena  of  form,  position  and  magnitude,  which  demand 
exj)lanatiou  as  do  other  ])hysical  ])henomena.  This  ex})lanation 
the  geometrician   gives  by  the  assuni])tion  of  certain   hypotheses, 

*('lifiiir(l,  \\'.  I\.,  "'A  I'reliininary  Sketcli  uf  lUciiniternions,"  MiitJu'inntirnl 
/•./y»/v.   Nn.    XX. 

Kk'iii.  v.,  "Ziir  Niclit-lMilcliilisclien  ( u'onit'trie,"  Moth.  Aitnulcn,  vol.  ."7 
(1S'.)(I).  ],.  :',14.  Lrrfirr,'.^  oil  Jfiiilu'ui'itin;,  Lvvuux'  XI.  ^L-w  Y(irk.  IS'.tl.  "Zur 
erstt'ii  \\'rtlu'iliiti,<:  dcr  Lohati-hcwsky  l*ivise,"  M((lh.  Amuthn.  vol.  o'.t  (ISUS), 
especial !\-  pp.  .llM-o'.fJ. 

KilliiiLf,  \\'.,  "  leher  ClilTord-Klein'sclie  Ivaunit'onuen,''  Math.  A  inxilni.  vol. 
o!)  (IS'.ll  1.  Juii/iihruiiij  ill  (lir  G ninilliiiji-n  drr  (loninirir,  vol.  1,  Clia]).  4  ;  I'ailiT- 
horii,    iS'.t:;. 

"^  Aitn<il.<  nf  ^[nlhrm(tlii:-<,  ser.  1',  vol.  .">  (  I'.td'il,  p.  71. 

81 


o'2  '1M1I-:    DOSroN    C()LL()<^l^IU.M. 

wliicli  lie  is  i'vvv  to  make  as  he  i)leasos,  provided  that  they  are 
self-('i»nsistent.  I'he  test  of"  the  validity  of  the  hv])()theses  lies  in 
th(,'ir  i-esults.  \\v  make  at  tirst  hy])otlieses  whieli  follow  the  ideas 
of"  Rieiiiann's  famous  JLi/ji/ifdJioiisschriff.'''' 

It  is  admitted  that  (questions  may  be  raised  whieh  lie  haek  of 
these  hvi)otheses,  as,  for  exam})le,  the  possibility  of  i'eduein<^  them 
to  simpler  axioms,  but  the  diseussion  of  such  (piestions  lies  outside 
our  j)resent  j)rovinee.  The  Iviemann  method  has  for  us  the  double 
advantage  of  allowing-  the  immediate  use  of  analytic  methods  and 
of  restricting  the  discussion  at  the  outset  to  a  small  region  of  space. 

A  geometry  having  thus  l)ecn  developed  in  a  restricted  ])ortion 
of  s])ace  is  extended  to  all  s[)acc  by  means  of  new  hypotheses, 
which  ai'e  essentially  those  used  by  Killing  in  his  (truiuUdijcii 
(h'r  (jcoinctric.  In  the  further  dcvelo])ment  the  ideas  of  the  last 
named  treatise  have  been  largely  followed. 

1.    Till-;   Fii'tsr  Two   II yi-othesks. 

As  already  said,  we  adoj)t  in  our  investigations  the  method  of 
Iviemann  bv  which  our  objective  sj)ace  is  assumed  to  be  an  example 
of  an  extent  { Ma iitiUj/dirikcil)  of  three  dimensions  in  wliicli  an 
element  may  be  determined  bv  means  of  coJn'dinatcs.  Wv  assert 
tlii-  (vxplieitlv  in  the  following  words  : 

I'^IIlsT  1 1  Vi'(  )TII  KSIS.  Sj)(ii'f  i.s  (I  cnnt'i  nil  II  III  of  tlinc  iliiiiiiis/on.s- 
III  ir/iicfi  (I  jiniiil  111(11/  1)1  ill  ic/'iiil  iiril  hif  llirci  / iiilejiC/ii/i  nf  lUn I  roi]/-- 
(hiiiilis  (:  ,: ,,  :,,).  //  <i  itrnncr/i/  r<'.-<l  iirh  il  nnrlinn  nf  sjnice  i.<  rmi- 
siilrrril,  llir  cnm  sitDiidriK-i'  hilircni  j/ainl  miil  coi'irdi  iinh  ;.v  oiic-ln-oite 
(I nil  I'mili  II  mills. 

\\  itiiiii  iiiii-  space,  we  m:iv  pick  out  at  pleasure  one-dimeiisioiial 
exteiil-  (ii-  line-.  We  sliall  restrict  ourselves  to  lines  which  may 
be   expre,--e(l    b\-    t  he   e(|liat  ioii-. 

:,-./■,(/),  ■-,=.fjt\  -:;=./';(0, 

whei'c  /  i-  an  arbitrai'\-  pai'anieter   and  ,/,,,/,,  and/',  are  cnniinuous 

'  I ;  h  iiM  ii:i,    II..    "I    c'hiM    (111'    1  1  y  I 'dl  lic'-cii,   wilclic    dii'   <  icniiict  :ic'    /u    (ii'iii'.ile 


FORMS   OF    X()X-Fl'CLII)EAN    8PACF.  ■'>■> 

functions  possessing;  continuous  first  dcrivatcs,  nowhere  vanisliinu; 
simultaneously.  For  such  a  line  we  may  introduce  the  concej)- 
tion  of  length  as  follows.  Consider  a  ])ortion  of  the  line  corre- 
spondini;  to  values  of  t  lyinu"  between  the  values  /„  and  T  inclu- 
sive, and  let  this  portion  be  divided  into  n  segments  to  the 
extremities  of  which  correspond  the  values  f^,  f.„  f.^,  ■  ■  ■  /  _j,  T.  Let 
further  ("p  r.„  z^)  and  (:■,  -f-  Sx^,  z.,  -f  8z.^,  z.^  -f-  hz.,)  be  the  coordi- 
nates of  the  extremities  of  any  segment,  corresponding  respect- 
ively to  t^  and  /.^,.      We  may  then  assume  arbitrarily  a  function 

4>(z^,z^,z^]   S^„  &,,  ^) 

which  has  the  following  two  pro])erties  :  First,  it  shall  become 
infinitesimal  with  hz^,  8z.^,  S:^,  and  consequently  with  t^_^  — ^  5  ^^^^^ 
secondly,  the  sum  of  the  /;  values  of  this  function,  computed  for  the 
n  segments  of  the  line,  shall  approach  a  limit  as  n  is  indefinitely 
increased  and  each  of  the  n  (juantities  /._,  —  f.  approaches  zero,  this 
limit  to  be  indej)endent  of  the  manner  in  which  the  segments  of 
the  line  are  taken.  lliis  limit  is  defited  as  the  length  of  t/ie  line. 
If  in  particular  we  take 

the  length  of  the  line  is  expressed  by  the  integral 


d.r.  (I 


^       a.c.  (i.t  ,    ,, 


The  differential  of  this  integral,  namely, 

we  call  the  liHe-elcinenf  of  the  space.  AVe  express  tiiese  conventions 
in  a  new  hyj)othesis  as  follows  : 

Skconj)   irvi'OTllHsis.      T/ie  lenr/tlt  of  a  line  shall  he  determined 
by  inea/is  of  a  line-eleinent  (jivea  hi/  the  equation 

irhen  the  a .^^  are  fnnetions  (f  z^,  z.,,  z.^,  posscssinf/  coiitin}i(nis  drriva- 
tiv(s()j  the  fird  Jon_r  orders,  the  determinant  j*"?,/. '  d()es  not  v<nu'sh 
idt  nlmid'ji,  and  flie  expression  imder  tJie  radical  si</n    is  positive  for 


34  THE    BOSTON   COLI.OQUIUM. 

all  ml II ex  ~,,  .:.„  Zy  dz^,  dz.,,  (Jz.^,  provided  fhdt  {z^,  z.„  z.^  h  a  point 
of  spore  and  fJud  dz^,  dz.„  dz^  are  not  all  zero. 

2.  Definitions. 

AVe  jirocoed  now  to  develop  the  conceptions  of  a  geodesic  snr- 
face,  a  geodesic  line,  an  angle  and  a  direction,  which  shall  corre- 
spond to  the  conce])tious  of  a  straight  line,  a  plane,  an  angle  and 
a  direction  in  P^uclidean  space. 

1 .  (u'ode-'iir  Line.  A  geodesic  line  is  defined  roughly  as  the 
shortest  distance  between  two  points.  To  determine  its  equa- 
tions, we  have  to  find  the  conditions  that  the  integral 


J,.  \^"<"  ,11  ,i 


U'" 


shall  be  a  minimum.     The  Calculus  of  Variations  gives  as  neces- 
sary conditions,  the  three  e(juations 

where  1=1,2,  -5,  and 

dz.  dz, 

''•=^-".^,H  ,ir 

If  we  take  as  the  independent  parameter   the  length  -•>?,  as  defined 
bv  the   integral,  these  equations  take  the  somewhat  sinq)ler  form 

dr    dz^        dz.,        dz.;^     ^^^■<'a/''-,'^-k 

d.s  [''"  d.s  "^  "'-  r/.s  "^  "'••'  ds\  ^  ^'^  Pz^^  d.^  ds' 
d  \       dz  dz,  '^■■^■tl        1^  ca    dz.  dZi 

d^'^^d.-^''-d:+"^d.\  =  ^  cz^di  </.' 

d[        dz^  dz.,  dz.^-^        ^^ca^,,dz^dz,. 

f/.s'l       '■  r/.s'  -'  a.s  •"   f/.s  J         -       t  :i.j    ils    (Is 

which  must  be  consi(h'i'ed  in  connection  with  the  identity 

dz.  dz 


^  a 


:    '■  =  \ 


d.^    ds 
("Diu'crsely  these  conditions  ai'e  sufiicient  if  ^'  is  not  too  great. 


FORMS    OF    NOX-KrcLIDEAN    SPACi:.  o-") 

Moi-e  precisely  :  I-^et  (z^',  z',^',  r^")  be  any  point  point  of  space,  and 
(:,,  z.„  z.,^  any  second  point  such  that  'z.  —  .:!*  j  does  not  exceed  a 
suitably  chosen  positive  quantity,  h.  Then  the  above  e(iuati()ns 
admit  one  and  only  one  solution  which  passes  through  the  points 
(: " )  and  (;)  and  has  all  its  points  lying  in  the  region  ' ,-'!''  —  z.'  <  A  ; 
aud  foi'  the  corresponding  curve  the  integral  s  has  a  smaller  value 
than  for  any  other  curve  joining  the  })oints  {z"^)  and  (c). 

We  take  the  equations  accordingly  as  the  defining  equations  of 
tiic  geodesic  lines  and  shall  ap})ly  this  name  to  the  curves  satis- 
fving  these  equations,  even  if  the  curves  have  l)een  so  prolonged 
that  the  minimum  property  no  longer  holds. 

2.  Direction.  In  accordance  with  the  theory  of  differential 
equations  it  is  always  possible  to  find  one  and  only  one  solution 
of  the  above  equations  which  takes  on  at  an  arbitrary  point  (z^,  z.,,  z^) 
any  arbitrary  values  (not  all  zero)  of  the  differential  coefficients 

(h^       <h^       <lz., 

<}h'       ,h'       <is' 

If  these  differential  coefficients  satisfy  initially  the  condition 

"'  (Is    <l.s 

this  relation  will  be  fulfilled  for  all  values  of  x. 

The  geodesic  lines  which  radiate  from  a  point  are  hence  dis- 
tinguished from  each  other  by  the  ratios  of  the  values  of  the  dif- 
fei-ential  coefficients,  which  may  consequently  be  regarded  as 
fixing  the  direction  of  the  line;  the  direction  being,  broadly,  that 
property  of  the  line  which  distinguishes  it  from  all  others  through 
the  same  point.  It  will  be  convenient  to  denote  (Jz.jds  by  ^.  and 
to  speak  shortly  of  the  direction  (^,,  ^„,  l^.^,  or  ^.  These  quantities 
satisfv  the  relation 

'■\.  Ait^/U.  The  angle  6  between  two  intersecting  curves  with 
the  directions  ^'  and  ^"  is  defined  by  the  equation 

cos  e  =  ^n^XX'r 


3(1  THI-:  r>osTox  colloquium. 

In  particular  two  intorseeting  curves  are  perpendicular  if 

4.  (rcodcfiic  Surtacf.  A  geodesic  surface  is  defined  as  a  pencil 
of  geodesic  lines.  More  precisely  :  Take  any  two  geodesic  lines 
()A  and  OB  intersecting  at  0,  having  at  that  point  the  directions 
a  and  (3  respectively,  and  making  the  angle  (o  with  each  other. 
Consider  any  other  geodesic  line  OJ/ with  the  direction 

where  X  and  fx  are  parameters  sul)ject  only  to  the  condition 
X-  +  /x"  +  2X/A  cos  ft)  =  1, 

which  arises  from  substitution  in 

As  X,  /u,  take  all  ])ossil)le  values,  OM  generates  a  pencil  of  lines, 
which  is  (U'fined  as  a  geodesic  surface.  It  may  he  shown  without 
diiliculty  that  in  this  ])encil  there  is  one  and  only  one  line  }>er- 
jx'udicular  to  OA  and  that  this  may  rei)lace  0/>  in  defining  the 
j)encil.      W'e  shall  then  have 

^.  =  a.  cos  0  +  /3,.  sin  6, 

whci'c  0  is  the  angh'  Ix-tween  OA  and  0.1/. 

If  now  /'  is  anv  point  on  OM  and  /•  is  the  length  of  ()!',  the 
(•(x'lrdinates  z^  of  /' ai'e  determined  hy  integrating  the  eijuations  of 
the  g(M)(Jcsic  lines,  choosing  the  .solution  which  has  at  O  the  direc- 
tion ^,  and  substituting  /■  for  x.      We  have  tiien 

-,=./•( r,,  'c.  t,  '•)  =  f(^,  /•'), 

the  I'liiictioii-  (/)^  being  continuous  together  with  their  pai'tial 
dci'I\at  i\e~  o)'  the  fir-t  and  >ccond  oi-(h'r^.  \\\'  taking  6  and  /•  as 
in<|i|>iiii|cnt  parameter.-,  we  ha\'e  the  e(|iiations  of  llie  geodesic 
-ni'tacc-. 


FORMS   OF    NOX-EUCLIDFAX    SPACE.  37 

o.   Till-:  TiiiUD  Hypothesis. 

The  method  of  superposition,  involving  the  assumption  that 
a  geometric  figure  may  be  moved  from  one  position  to  another 
without  altering  its  size  or  properties,  is  fundamental  in  the 
P^uelidean  geometry  and  would  seem  to  be  a  necessity  in  any  ex- 
planation of  spatial  ])henomcna.  The  hypotheses  thus  far  made 
do  not  carry  with  them  the  necessity  of  any  such  superposition. 
This  may  be  clearly  seen  by  examples  from  the  Euclidean  geom- 
etry of  a  kind  which  we  shall  frequently  employ  in  the  follow- 
ing j)ages.  In  thus  using  the  Euclidean  geomety,  we  do  not  as- 
sume that  it  is  objectively  true,  but  that  it  is  a  self-consistent  system 
which  exjilains  experience.  Consider  any  surface  on  which  a  sys- 
tem of  curvilinear  coordinates  {n,  v)  have  been  established.  This 
surface  is  a  two-dimensional  space  satisfying  the  first  two  hypoth- 
eses, the  line  element  being  of  the  form 

r/,s-  =  E(hi-  +  -IFdujlv  +  Gdr. 

Such  surfaces,  however,  offer  various  ])ossibilities  in  the  matter  of 
•sujierposing  one  portion  upon  another.  One  needs  only  to  con- 
sider the  elli})soid,  the  right  circular  cylinder,  and  the  sphere  as 
examples. 

To  bring  the  principle  of  su[)erposition  into  our  present  discus- 
sion, we  shall  define  a  displacement  as  a  transformation  by  which 
a  continuous  portion  of  s})ace  is  brought  into  a  continuous  point 
for  ])oint  correspondence  either  with  itself  or  with  another  portion 
of  space  in  such  a  manner  that  the  lengths  of  corresponding  por- 
tions of  lines  are  the  same.  Let  .V  be  a  portion  of  space  in  which 
the  coi'trdinates  of  a  point  P  are  (z,,  z.„  z.^),  and  let  S'  be  a  portion 
of  space  in  which  the  coordinates  at  a  point  P '  are  (z[,  z'.,,  z'^). 
Let  the  line-element  in  .She  denoted  by 

(J.s  =  1   la.jlz.dz, 

and  the  line  element  in  -S  '  by 

d.s'  =  1   y.(([^dz[dzl, 


•38  THE   BOSTON   COLLOQUIUM. 

wliere  (i'.^^  denotes  the  value  of  a.^.  for  (z[,  z'^,  z'^).  In  order  that  S 
may  be  displaced  into  -S",  it  is  necessary  that 

by  virtue  of  relations  of  the  form 

K   =  ylr  (■y'     t'     y'\ 
'■'i  —   r  A"i'  '^2'  "3/' 

where  the  yjr.  are  continuous  functions  of  (z[,  s.',  z'^),  possessing 
continuous  first  derivatives,  and  establisning  a  one-to-one  relation 
between  the  points  of  *S'  and  ^S". 

ft  is  easy  to  show  that  by  any  displacement,  geodesic  lines  are 
transf  )rnied  into  geodesic  lines,  geodesic  surfaces  into  geodesic 
surftices,  and  angles  are  left  unchanged. 

The  existence  of  displacements  in  space  is  made  the  subject  of 
a  new  hypothesis. 

Third  Hypothesis.  If  P  is  any  point  of  sjjace,  it  shall  be 
possible  to  displace  a  restricted  portion  of  space  surrounding  P  upon 
itself  ill  such  a  manner  tlud  any  two  geodesic  lines  through  P  shall 
correspond  to  any  oilier  tiro  geodesic  lines  through  P,  provided,  only 
that  the  tiro  Udter  lines  make  the  same  angle  with  each  other  as  do 
the  tiro  former  lines. 

The  (juestion  of  disj)lacement  of  a  surface  is  intimately  con- 
nected with  the  (jjuantity  called  by  Gauss  the  measure  of  the 
(Mirvature,  or  simply  the  curvature,  of  the  surface,  lender  that 
term  we  understand  a  (juantity  7\' defined  ))y  the  relation 

1  /  r   r  /•'  cE  1  rf,n 
'  ~  2,     AY,'  _  r-\cii  [  /';,     jrj;  _  //2  t^,.  ~  ,     y.y;  _  yi   Cu  J 

2  cF  1  CE  E  rA'1\ 

,    AY/  _  /.'-  Cii  ~  ,    /-Y/  _  ir^  cv  ~  A',    /.;<;  _  f"-  c,i  ]) 

With  the  geometric  interpretation  of  the  curvature  as  usually 
given  on  the  hy])otIiesis  that  the  surfae(»  lies  in  l']uelidean  space 
we  have  nothing  to  (1(».  J"'or  ns  the  eiirvat  lire  is  siinplv  the  above 
e\pi'es>i()ii  which  is  fully  ileteniiined  when  the  liue-elcnient  of  the 
surface  is  t:i\-e!i,  and  may  be  shown  to  be  an  invariant  of  the  sur- 
face, that   is  imlcpeiident  of"  tiie  co("u'dinat(>s  used    to  define  a  point 


FOKMS    OF   XOX-EUCLIDEAN    SPACE.  -UJ 

upon  tlie  surface.  When  K  is  the  same  for  all  points  of  the  sur- 
face, the  surface  is  said  to  be  one  of  constant  curvature.  Tiie  im- 
portance of  the  curvature  lies  in  the  two  theorems  : 

.1  necessary  CO iidif ion  that  two  portions  of  snrf((ces  may  he  hrouglit 
into  point  for  point  correspondence  iritli  presercation  of  distance  is 
that  they  have  the  same  ciircature  (d  corresjjondi)i(j  points. 

If  the  tiro  portions  of  surfaces  are  of  constant  curvature,  the  con- 
dition is  (dso  sufficient. 

The  Gaussian  measure  of  curvature  of  a  surface  is  extended  by 
lliemann  to  s})ace  of  //  dimensions.  For  three  dimensions  consider 
a  point  {z^,  z.„  .^.,)  and  two  directions  (-x^,  a.,,  a^)  and  (/S^,  /3.„  /3.^), 
taken  from  that  })oint.     Then  the  Riemann  curvature  is  a  function 

K{z^,  z.,,  z.^;   a^,  a.^,  a^  ;   ^„  fS.,,  /S.J 

which  ij^ives  the  Gaussian  curvature  of  tiie  geodesic  surface  deter- 
mined 1)}'  the  point  and  the  directions.  The  Riemann  curvature 
of  a  general  space  is  accordingly  dependent  both  on  the  point 
of  space  for  which  it  is  reckoned  and  on  the  directions  of  the 
lines  taken  through  that  point  to  define  a  geodesic  surface.  I^ut 
if  the  space  satisfies  our  third  hypothesis,  the  curvature  is  a  func- 
tion of  the  point  only.  For  by  this  hypothesis,  any  two  geodesic 
pencils  with  their  vertices  at  the  same  point  P  may  be  brought 
into  point  for  j)oint  corresj)ondence  with  preservation  of  distance. 
Hence  by  the  surface  theorems  above  quoted,  the  two  geodesic 
surfaces  formed  by  the  pencils  must  have  the  same  curvature  at 
corresponding  points  and  in  particular  at  1\  Schur  *  has  proved 
that  when  the  curvature  is  thus  constant  at  each  j)oint,  it  does  not 
change  as  we  pass  from  })oint  to  point.  The  space  is  then  said  to 
be  of  constant  curvature.  A  new  ]>roof  of  Schur's  theorem  will 
be  given  in  the  following  paragra])h. 

4.  Tin-;  Link-Element. 
Take  any  point  O  at   wiiicli   the  functions  r?.^  arc  single-valued 
and   continuous.      Then,  as   we   have  seen,  there  exists  around  O 

'"Schur,  1'.,  "  Uel)er  dt'ii  Ziisjunnii'iiliant,''  der  Ivuinic  I'dnstanien  liieinann'- 
sclien  Kriiiiiiiiur.gsinasses,"'  M<tiJi.  Anniil<'ii,  vol.  liT  (ISSd),  p.  51'.'!. 


40  THE    BOSTON    COLLOQUIU^r. 

a  region  of  s])ace  such  tliat  any  point  P  of  tlie  region  can  be  joined 
to  0  by  one  and  only  one  geodesic  line  lying  in  the  region.  We 
shall  call  this  region  T.  Through  0  take  in  T  three  mutually 
l)erpendicular  geodesic  lines  OA,  OJl,  OC.  This  can  be  done  l)y 
taking  directions  (otj,  o:.„  a.,),  (/3^,  /3.„  /3,),  (7^,  7.,,  7,)  so  as  to  satisfy 
the  relations 

^a':,-y.B,^  =  0,     ^n:]:^r^,  =  0,      2a-7^^,  =  0, 

where  f/ .'-''  si»;nifies  the  value  of  <(.,  at  0.  The  direction  of  any 
geodesic  line  through  0  is  then 

where  '/,,  d.,,  a.,  are  in(h']KMident  ])arameters  subject  only  to  the 

condition 

''1  +  "I  +  "1=  '^y 
which  arises  from 

The  dir('(;tion  may  accordingly  be  named  by  means  of  (c/^,  a.,,  a^). 
Let  P  l)e  any  jioint  on  this  gcotlesic  line  and  let  the  distance 
01'  \h'.  denoted  by  /•,  where  /•  is  ])ositive  if  measured  in  the  direc- 
tion '/^,  and  negative  if  measured  in  the  opposite  direction.  We 
may  take  the  (piantities  (rr,,  c/.,,  a.,,  r)  as  the  coordinates  of  I\ 
Then  to  any  set  of  values  of  the  corii-dinates  corresponds  only  one 
]»oint  l\  and  to  anv  point  /*  correspond  onlv  the  cor>rdinates 
{('^,  ",,,  ir.,,  /■)  or  (—  (f^,  —  (i.„  —  (1.^,  —  /•).  J^x'tween  old  the  and 
new  coiu'diuates,  there  e\i.-t  relations  ol'the  f»rm 

•-,  =  /■',(/' I'  ''.•?  ",'  '■)' 
where    the    functions    /•'   ai'c    continuous    and    |)ossess   continuous 
(lerivali\e-  i<['  the  tii'st    two  orders  since   they  are   tlie  solutions  of 
the  ditfereiitial  e(jiiati(>n>  ol'the  geodesic  lines. 
Wy  the  >ubstitution  in 

tlie  t'liriii  of  th<'  iiiie-eh'ment  is  olMained  as 


FORMS    OF   NOX-ia'CLlDFAN    SPACE.  41 

i/c  I 

wliere 

cz.  P 


Tlie  (liroct  calculation  of  the  values  of  the  coeflficieuts  is  difficult : 
but  we  shall  prove  by  an  indirect  method  that  the  proper  form  is 

where  k  is  a  constant. 

To  do  this,  consider  any  curve  (',  defined  by  the  equations 

a,=^J\{t),     a,=flt),      o.^m,      r=j\{t). 

If  7*j  is  any  fixed  point  on  ('  and  6  is  the  angle  between  the 
geodesic  lines  OP,^  and  OP,  ^  is  a  function  of  /  and  hence  t  is  a 
function  of  0,  which  for  small  portions  of  C  is  one  valued.  We 
may  consequently  write  for  the  equations  of  G 

If  from  these  four  equations  we  omit  the  fourth,  thus  allowing 
)■  to  take  any  value,  we  have  the  equations  of  a  surface,  which 
passes  through  the  curve  C,  as  is  evident,  and  also  contains  the 
point  0  since  the  equations  are  satisfied  by  /•  =  0.  The  surface 
is  analogous  to  a  cone  of  the  Euclidean  geometry,  for  the  lines 
0  =  const,  are  geodesic  lines  radiating  from  0  to  the  points  of  C. 
These  lines  form  one  of  the  systems  of  coordinate  curves  on  the 
surface  ;  the  other  system  is  composed  of  the  lines  /•  =  const., 
each  of  which  is  the  locus  of  jxjints  equally  distant  from  0.  If  we 
refer  to  the  general  form  of  the  line-element  of  a  surface 

r/.r  =  ]'j!r  +  '2F(Jr<ld  -f  ('d6-, 
it    is    clear   that    in    the   present  case,   P^  1,  since  .s' =  /■  when 
^  =  const. ;     and    /*'=(),    since    the   curves    r  =  const,    cut    the 
geodesies  6  =  const,  at  right  angles  by  a  theorem  of  the  Calculus 
of  Variations.*      We  have  therefore  on  the  surface 

*  Cf.  Kneser,  A.,  V(iri<iJion.<rrch)iini;/,  ji.  JS.  ISol/.a,  ().,  ChIcuIuh  nf  Variation.^i, 
p.  Ui4. 


42  THE    BOSTON    COLLOfiUIUM. 

and  we  proceed  next  to  replace  (W  by  its  value  in  terms  of  «.. 
For  that,  we  call  hO  the  ani^le  between  two  neighboring  geodesic 
lines  OP  and  ()(2,  with  directions  a  and  (i  -f  hd,  where 

«\  4-  ^K  +  (q  =  1, 

Then 

cos  86  =  ((^(((^  4-  Sa,)  +  a.J^".,  4-S^'.)  +  '''./"s  +^'';i) 

=   1  -f  '^jSr^i  +  (i.,8((„  -f  ^'3Sc/.j 

=.  1  _  i(8al  +  grr^  +  8(q), 
so  that 

sin-  .^    =  :'(6(f-  +  6<q  +  da"). 
From  this  follows  in  the  differential  notation 

So  that  the  line-element  of  the  suface  is 

^/,s--  =  G{(h{\  +  (hr,  4-  (hr^)  +  ^?/''. 

This  is  in  particular  the  element  of  the  length  of  the  curve  (\ 
since  f '  is  on  the  surface.  ]^>ut  C  is  any  curve  in  space  and  hence 
the  above  exj)rcssion  is  the  line-element  of  the  space. 

W(>  seek  now  to  determine  (i.      For  that  purj)()se  consider 

cz    Cz , 

where  (see  ]).    10) 

'■,^   ^'i<"i'  "j'  ":;'   '■)  =  '■'!"  +  ("i^,  +  ('2/^.  +  ";i7,)''  +    "  '  '  • 

Ilenee 

(;  =  rln^^^^^^^  +  ■■■  . 
and  e()n-;e(|iieiitlv 

(1  (•),  „=  <»,        (     ^ 

Tim-  I'ar  the  di<en~<i(in  !■-  applicable  to  aiiv  s|)aee  which  satisfies 
till'  lir-t  two  livp(itlif~r-.  W'c  exaniiiie  now  tli(^  etVeet  of  intro- 
diii'iiiij-    the    thii'd    h\|iot  lir-c-;.        A    '■■eodcsie  surfiee    fornie(l    bv    a 


F0113IS  OF   XON-EUCLIDEAN    SPACJ:. 


43 


pencil  of  lines  with  its  vertex  at  O  is  a  special  case  of  the  conical 
surfaces  just  discussed  and  its  line-element  is  therefore 

ih-  =  Udd-  +  (//■-. 

The  formula  for  the  curvature  K  reduces  to 

1    cr\/G 

K  =  ^  .,    ' 

V  G    <■'" 

By  the  third  hypothesis,  any  one  of  these  surfaces  may  be 
brought  into  correspondence  with  any  other  by  means  of  a  dis- 
placement by  which  a  point  at  the  distance  r  from  0  on  the  one 
surface  corresponds  to  any  point  at  the  same  distance  /■  from  0  on 
the  other  surface.  Hence  the  curvature  K'ls  a  function  of /•  alone, 
that  is 

1    c^W 


'G     C 


;  =^('-)- 


From  this  and  the  conditions  governing  (/  when  /•  =  0,  it  follows 
that  G  is  a  function  of  /•  only. 

The  exact  form  of  G  is  obtained  by  the  following  considerations  : 
The  equations  of  the  geodesic  lines  in  the  new  coordinates  are 


(I'd.  (Id.dr 

G       '  +  2  6'       '  .    +Xa.=  0, 


('=1,'^'>) 


<Fr 


(;' 


ils-   ^  '1(1 


x  =  0, 


where 


\  =  <; 


G' 

= 

<1G 
dr 

'hi.;\ 

"4 

J' 

■::^(i)]-'-(:;;y 


Take   now  the  geodesic  surface  a.^  =  0,  for   which   the  line-ele- 
nient  is 

<Js-  =  Gi(h(]  +  (hr)  +  dr, 


and  a[)ply  the  Calculus  of  \'ariations  to  determine  the   shortest 


44  THE   BOSTON   COLLOQUIUM. 

lino  on  this  snrface  connecting  any  two  points.  Sucli  a  line  exists 
if"  the  j)oints  are  not  too  remote,  and  its  etjuations  will  be  found  to 
he  exactly  those  obtained  when  a,^  is  ]>laced  equal  to  0  in  the  equa- 
tions of  the  geodesic  lines  in  space.  It  follows  that  any  two  points 
on  the  surface  a.^  =  0  may  be  connected  by  a  geodesic  line  lying 
wholly  on  the  surface.  In  ])articular  any  point  of  the  surface  is 
the  vertex  of  a  ])encil  of  geodesic  lines  which  lies  on  the  surface. 
Take  n<nv  ]\  any  point  in  a.,  =  0,  and  clioose  on  the  geodesic 
line  O/^i  the  [)oint  J/ equidistant  from  0  and  J\.  This  point  31 
may  be  used  as  the  vertex  of  a  pencil  which  covers  the  surface. 
By  the  third  hypothesis,  there  exists  a  displacement  by  which  this 
pencil  is  self-corresponding,  the  point  Jf  being  fixed  and  the  geo- 
desic line  MP  corresponding  to  J/0.  Hence  the  curvature  of 
r/,^  —  0  at  1\  e([uals  that  at  0,  and  the  surface  is  consequently  one 
of  constant  curvature.  But  the  surface  (/„  =  0  may  be  brought 
into  corres[)ondence  with  any  other  geodesic  surface  formed  by  a 
pencil  of  lines  with  vertex  O.  Ilencc  K  is  inde{)endent  of  >• 
throughout  and  is  conse({uently  constant.     AVe  ])lace 

and  have  the  three  cases  of  a  s])ace  of  constant  j)Ositive  curvature, 
a  ^pacc  of  constant  negative  ciu'vature,  or  a  space  of  zero  curva- 
ture, according  as  /:  is  real,  pure  imaginary,  or  zero.  To  determine 
(i,  wv  have  the  differential  e([uation 


1    (^ 

with  the  initial  conditions 


i'r' 


(I    ^'),    ,„       {      ^        ]       =1- 


I  Icncc 

1 


(; 


un 


/,-/• 


if/,-  i-^  real  tliis  (h'tcnnination  df /.•  is  liiial. 


1  (^ 


FORMS   OF    XOX-FrCLIDKAN    SPACE.  45 

If  /;  is  ]nire  imauinary,AVC  may  jilace  /;  =  I/:'  and 

sinh  /.•'/■ 

If  k  is  zero,  we  may  })lace 

sin  l:r 
^    f;  =  lAin  -  J  -  =  /'. 

It  will  l)e  more  convenient  to  retain  the  general  form  for  /;,  since 
the  above  changes  are  readily  made.  We  have  accordincrlv  the 
line-element  in  tlie  desired  form, 

(^■^-  =  '     ,,,  "'  i'^n]  +  (Jfil  -i-  (!((l)  4-  (lr\ 

It  is  to  be  emphasized  that  we  have  shown  the  existence  of  a 
displacement  by  which  0  is  transferred  into  any  other  point  P^ 
and  reciprocally.  By  the  combination  of  two  such  dis})lacements, 
a  dis))lacement  may  be  found  l)y  which  any  point  7''^  of  T  may  be 
made  to  correspond  to  any  other  point  P^  of  T. 

5.    Geomktuy  IX  A  Restrictkd  P()P>tiox  of  Space. 
AVe  shall,  for  the  present,  confine  our  attention  to  the  portion 
of  space  /'already  defined  and  introduce  the  coordinates  * 

sin  ^v  ,  ;      C^) 

•^■  =  ^  -/.  -'  ('  =  1,  2,  3)  I 


where 


l  +  lr(.rl  +  .rl  +  .rD=l.  (2) 

The  line-element  is  now 

.^N-  =  j^''.'i  +  dc]  +  >Jj-:  +  (h-:,  (3) 

and  the  diiferential  e(|Uations  of  the  geodesic  lines  are 

\^^+/r.r^  =  ().  (7=0,1,2,;])      (4) 

*'riK'se  coiirdiiKites  arc  eallod   hy  Ki!liii<T  tlic  "\\'eii'i>tra>sian  (•o(lI■(linate^,  \k'- 
cau^e  tliev  wci-e  lir-t  used  1)V  AVeicrslrass  in  seminar  wovk  in  1<S7l*. 


4(5  thp:  BOSTON  colloquiu:m. 

The  integrals  of  these  equations  are 

X.  =  A.  sin  Is  +  B.  cos  h,  (i  =  0,  1,  2,  3)    (5) 

wiiere  the  constants  must  be  so  chosen  as  to  satisfy  the  conditions 

Al-^^A;  +  Al  +  Ai)=l,^ 
Bl  +  1^{B\  +  m  +  BD  =  1,  (6) 

A^^B^  +  h\A,B,  +  A^B^  +  A.^B^)  =  0,  J 

which  are  necessary  and  sufficient  in  order  that  the  conditions  (2) 
and  (o)  may  be  satisfied.  In  fact  the  constants  i?.  are  the  codrdi- 
nates  of  the  point  from  which  s  is  measured  and  the  constants  ^^1 . 
are  the  values  of  dx.jds  at  that  point  and  consequently  fix  the 
direction  of  the  line. 

We  may  write  the  equations  of  a  geodesic  line  in  terms  of  any 
two  points  upon  it.  Let  y.  and  ?;.  be  the  two  points,  and  let  I  be 
the  distance  between  them  measured  on  the  geodesic  line.  If  we 
measure  .v  from  z.,  we  have  from  (5), 

a  =  B.,     7/.  =  A.  sin  Id  -f  B.  cos  hi  (7) 

Vvova  these  follow,  with  aid  of  the  relation  (G), 

an  important  formula  which  gives  the  distance  between  two 
points  in   T. 

If  oi\  is  anv  other  ))<)int  on  the  geodesic  line,  we  have  from  (o) 

and  (7) 

:,r  ^  X,/.  +  /x,-i_.,  (9) 

where 

sin  /:s  sin  /;(/  —  .s) 

sm  Id  sin  ki 

or,  otherwise  wi'itten, 

A.  sin  /•/  =  sin  ks,      \  cos  hi  -\-  ^  =.  cos  hs. 

Hence  A.  and  /u,  nnist  satisfy  the  condition 

X-  4-  fx'  +  2X/X  cos  /•/=  1,  (10) 


FOmr^    OF    NOX-EI'CLIDKAN    SPACE.  47 

"svhicli  is  also  the  necessary  and  sufficient  condition   tliat   :i\  may 

satisfy  i-elation  ('2). 

Convei'-ely  any  equations  of  tlie  form  (9)  for  whicli  conditions 

(10)  and  (S)  hold  represent  a  geodesic  line,  provided  they  are  sat- 

i.-tied  by  points  in  7'.     For  it  is  always  })ossil)le  to  find  an  angle  v, 

such  that 

sin  kv  =  A.  sin  /,•/, 

cos  J:p  =  A.  cos  /,■/  +  /x. 

From  the  condition  (•">)  it  follows   that  da- =  (h>-.      It  can  then  be 
verified  that  the  functions 

''\  =  ^y.  +  ^S 

satisfy  the  ditferential  equations  (4). 

We  collect  these  important  results  in  tiie  following  theorem  : 
Any  (jeofh's'tc  line  inaij  he  luprejtented  hji  ilie  rqiiafhrns 

•'\  =  ^!/,  +  /^^,,  ('=  0,  1,  2,  3) 

irjit'/'f  _//.   ((ii(f  X.  (ire  cnij  firo  points  on  fJie   line,  (in<}  X  and  /x  (/re 
jHtiriinefery  safisfi/inr/  fJie  reledioii 

X-  -f-  /Li-  -(-  -iXfx  cos  /;/  =  1  , 

/  hein;/  flie  di.sfa ni-e  befireen  the  tn-o  jtoints  //.  (ind  z.. 

Conrcr.sc/i/  (iiitj  eqiKdions  of  the  above  form  )'epre.-<ent  a  f/eodesic 
line  if  tlicti  are  satisfied  hi/  points  of  T. 

Fi'om  this  follows  immediately  : 

Aioj  tir()  tiuear  liOuiogeneons  e(ju<(fions  in  .''.  repre.-^ent  a  (/eodesir 
line  if  satisfied  hy  coordinates  of  points  in  T ;  and  conrersehj  any 
(/eodesir  line  may  he  represented  by  two  such  erpiedions. 

As  to  the  geodesic  surfaces  we  have  the  theorem  : 

xlny  r/codesic  sn^iface  is  represented  by  a  linear  hornof/eneons  e<pia- 
tion  in  :r.  ,•  (iiid  conversely  any  sitch  etpiafion  represents  a  f/eodesic 
svrfacr  if  if  is  satisfied  by  points  in  T. 

To  ])rove  the  last  theorem,  consider  a  pencil  of  geodesic  lines 
determined  by  two  lines  through  7?.  with  the  directions  A',  and 
A[  respectively.     It  has  the  equations 

:>\  =  {\A\  +  iiA'l)  cos  hs  -\-  B.  sin  hs, 


48  TflE    BOSTON    COLLOC^rir^L 

where 

X~  +  /J,"  -f  2X/X  cos  ^  =  1 

0  beiii":  the  angle  between  the  two  lines  A',  and  A"..  From  this 
it  readily  follows  that  the  coordinates  of  any  point  on  the  pencil 
satisfy  an  equation  of  the  form 

Conversely  if  this  equation  is  given  and  y.  and  z.  are  any  two 
points  satisfying  it,  the  point 

X.  =  \i/.  -f  /jiz.,  (X"  +  yu^  -f-  2\/x  cos  kl  =  1) 

will  also  satisfy  it.  Hence  any  points  on  the  locus  of  the  equa- 
tion may  be  connected  by  a  geodesic  line  lying  wdiolly  on  the 
locus.  The  locus  may  therefore  be  considered  as  a  pencil  of  geo- 
desic lines  and  is  therefore  a  geodesic  surface. 

Explicit  formulas  for  the  dis})lacements  in  7' may  now  be  writ- 
ten. Since  these  displacements  are  continuous,  one-to-one  point 
transformations  by  which  a  geodesic  line  is  transformed  into  a  geo- 
desic line  and  the  expression  for  cos  kl  is  invariant  they  wuU  have 
the  form  : 

K  =  7r'",  +  7,0",  4-  Xr'";i  +  7o-'"o^ 
where 

8;    f  /-^(a^  +  Z^^-f  7;)  =/r,  {/:=},  2,^) 

SB^  +  /r(nc^^^^  +  (ifi,  4-  7,7.)  =  <»•         (',  /'  =  <>,  1 ,  •-^,  -  ;   '  +  /') 

l''rom  these  conditions  it  follows  that  determinant  |  aj3.,y.^S^^ '  =zb  1 . 
If  we  add  to  oiii'  detinitioii  of  a  displacement  the  condition  that 
it  may  be  I'educed  to  the  identical  substitution  bv  a  continuous 
I'liauge  of  the  coeHicieiits,  we  shall  have  the  new  condition 


FOUMS    OF    XOX    FrC'LII)i:AX    SI'ACK.  4i» 

(  onver^ely,  any  linear  snhstitntion  in  wliicli  the  eoefficients  sat- 
isfy the  above  conditions  ro})resents  a  disphicenient  in  7',  ])i'ovi(led 
that  it  is  satisfied  by  at  least  one  ])air  of  corresponding  points 
in  T. 

We  have  now  the  fnll  data  for  constructing  a  system  of  geom- 
etry in  T.  The  following  are  some  of  the  fundamental  theorems 
which  are  readily  proved. ''  In  fact  some  have  already  been 
})roved  in  the  j)receding  discussions  and  the  theoi'ems  are  repeated 
here  for  completeness. 

].  A  geodesic  line  is  completely  and  uniquely  determined  by 
any  two  points. 

'1.  A  geodesic  surface  is  eom]>l(>tely  and  uniquely  determined 
by  any  three  points  not  in  the  same  geodesic  line. 

3.  If  two  ]>oints  on  a  geodesic  surface  are  connected  by  a  geo- 
desic line,  the  line  lies  wholly  on  the  surface. 

-I.  Two  geodesic  lines,  or  a  geodesic  line  and  a  geodesic  surfiice, 
intei'sect  in  at  most  one  point. 

o.  Two  geodesic  surfaces  intersect  in  a  geodesic  line,  if  they 
inters(>ct  at  all. 

<).  On  a  given  geodesic  sui'face,  one  and  oidy  one  geodesic  line 
can  be  drawn  })erpendicular  to  a  given  geodesic  line  at  a  given 
}K)int. 

7.  If  a  geodesic  line  is  jierpendicular  to  each  of  two  intersecting 
geodesic  lines  at  their  point  of  intersection,  it  is  pei'j>endicular  to 
every  line  of  the  ])encil  defined  by  the  two  intei'secting  lines. 

Such  a  line  is  said  to  be  j)erj)endicular  to  the  geodesic  surface 
defined  by  the  pencil. 

s.  Through  any  point  of  a  geodesic  surface,  one  and  only  one 
geodesic  line  can  be  drawn  perpendicular  to  the  surface. 

!),  Through  a  given  point  on  a  geodesic  surface,  one  geodesic 
line  can  in  general  be  drawn  jierpendicular  to  a  given  geodesic  line 
on  the  surface  not  passing  through  the  given  point,  anil  never 
more  than  one. 

10.   Through   a    given    jioint    not   on    a    geodesic   surface,   one 

'■'''  Pi'Dol's  of  all  these  tlieoreins  nia\-  lie  fomul  in  the  .  I /i/(((/s  article  already  cited. 


'JO  the    BOSTON    ('OLL0(H'ir-^r. 

geodesic  line  can  in  general  be  drawn  perpendicular  to  the  surface, 
and  never  more  than  one. 

11.  The  sum  of  the  angles  of  a  triangle  formed  by  three  inter- 
secting geodesic;  lines  is  etpial  to,  greater  than,  or  less  than,  vr, 
according  as  /;  is  zero,  real,  or  pure  imaginary. 

It  apj)ears  that  the  geodesic  lines  in  T  have  all  the  properties 
of  the  straight  lines  of  practical  life  or  of  the  Euclidean  geometry. 
In  the  endeavor  to  construct  a  material  line  which  shall  be 
"  straight,"  we  may  proceed  by  attempting  to  realize  the  shortest 
distance  between  two  points  by  stretching  a  string  or  otherwise. 
The  result  is  simply  a  geodesic  line  by  detinition.  Or  we  may 
look  for  a  line  which  may  be  revolved  upon  itself  when  two 
})oints  are  fixed.  This  is  also  a  property  of  the  geodesic  lines.  A 
geodesic  surface  has  the  properties  of  a  plane.  The  practical 
testing  of  a  ])lane  surface  by  the  application  of  a  straight  edge  has 
its  full  significance  in  T.  The  ])ractical  measurement  of  length 
and  angle  by  the  aj)[)lication  of  an  assumed  unit  is  also  possible 
in  1\  We  see  then  that  the  groundwork  of  ex})erimental  geometry 
is  the  same  for  all  s})accs  which  satisfy  our  three  hypotheses. 
These  sj)aces  agree  also  in  the  first  ten  theorems  above  stated. 
A  distinction  ajijK'ars  first  in  the  eleventh  theorem,  which  ap])ears 
to  ])rt'sent  a  means  for  determining  the  curvature  of  our  objective 
space.  The  test  fails,  however,  owing  to  the  imjiossibility  of 
exact  measurements.  All  we  can  discover  is  that  the  sum  of 
the  angles  D'i  a  ti'iangle  does  not  diller  very  nuich  from  ir  and 
it  is  ])()ssible  to  >how  that  if  the  sides  of  a  triangle  are  sufticiently 
large  compared  with  /.'  tlie  divergence  of  the  sum  of  its  angles 
from  TT  is  within  the  limits  of  the  errors  of  ol)sei"vation.* 

\\  (.'  ma\'  sav  then:  ^\iui  spacf  ii-liidi  salisfwx  f/ic  t/n-cc  /i ifijof/ifscs 
/>■,  (IS  j(t,'  (IS  iiiir  j)i-is(  tit  l:ii()irli-(1(/e  (/ots^  ni  Jn//  acco/'d  ir'itlt  all  /(wf-s 
(ij  r.rj)r/'inii'«\,  /tror/dcd  siiiliihlr  rajiics  di'c  (/ircii  to  the  coiistd iit-s 
I  Ill-Ill  r(  it, 

*  Sec,  t'l  ir  cxMiiiiilc.  the  (■;!  leu  l:ili  1)11  in  I  ,iili;iclicv>k  v"s  Xni  i  i/i  niiii-lrl.<rlii'  Althiiiiil- 
Imnfii.   liaii-laliMl  l.y   !•'.   Imii^'cI,    1 .1'iii/.ii:.    \^W.   \<\k   --   "J  1. 


FOKMS    OF    NOX-KUCLIDKAX    SPACE.  51 

*).    The  ForuTH  and   FriTii   IIvpotiiksks, 

In  order  to  extend  our  system  of  geometry  outside  of  the  region 
7',  new  hypotheses  arc  necessary.  Tiiese  hypotheses  must  be 
such  that  their  verification  transcends  experience,  but  it  lies  close 
at  hand  to  assume  that  certain  properties  whicii  are  true  as  far  as 
experience  extends  are  everywhere  true.  We  accordingly  frame 
our  hypotheses  as  follows  : 

Forirni  Hyi'()tiik.sis.  Aui/  portion  of  space  in  vhich  fJie 
(jrealc-^f  f/('o<h?.sir  dist'ince  does  not  exceed  soiae  constant  J/,  dependent 
on  the  nature  of  tite  space,  may  l)e  so  displaced  tliat  an  artjitrarij  point 
of  this  portion  of  .space  ma ij  be  made  to  coincide  with  any  point 
vhat(  rcr  in  sjiace. 

1"  IF  Til  Hypothesis,  .1  displacement  of  a  portion  of  space  is 
coiiij>lrt</j/  and  uni(piely  determined  by  the  displacement  of  any  por- 
tion if  sjxicc  n-hich  forms  a  three-dimensional  pewt  of  tjw  first  portion. 

The  meaning  of  the  fourth  hypothesis  may  be  illustrated  by 
the  phuie  and  the  cone  of  the  Fuclidean  geometry,  as  examples  of 
two  dimensional  spaces  satisfying  the  first  three  hypotheses.  The 
region  corres])onding  to  T  may  l)e  taken  indefinite  in  extent  in 
the  case  of  the  ))lane,  l)ut  for  the  cone  must  be  so  taken  that  no 
point  of  the  cone  shall  be  covered  more  than  once.  The  size  of 
thi-  region  on  the  cone  depends  then  upon  its  nearness  to  the 
vertex  of  the  cone.  It  is  clear  that  the  cone  does  not  satisfy  the 
fourth  hypothesis,  since  l)y  definition  a  dis])lacement  demands  a 
one-to-one  correspondence  of  two  regions  and  no  matter  how  small 
a  region  may  l)c  taken  on  a  cone  this  region  can  not  be  moved 
indetinitely  near  the  vertex  of  the  cone  without  overla])ping  itself. 
A  I'ight  eli'cidar  cylinder  in  Fuclidean  space  would  satisfy  the 
fourth  hy})othesis,  the  (juantity  M  being  then  the  circumference 
of  the  right  section.  Similarly  a  Fuclidean  sphere  satisfies  the 
fi)urth    hyj)othcsis. 

In  like  manner  the  fourth  hypothesis  applied  to  a  three  dimen- 
sional sjiace  rules  out  singular  points  and  involves  the  assumption 
that  sj)ace  is  boundless.  It  does  not  however  assert  that  space  is 
infinite  in  any  or  all  directions. 


52  THE    BOSTON    COLLOQUIl^M. 

The  fiftli  hypothesis  asserts  that  if  a  definite  displacement  is 
applied  to  a  region  of  space  S.,  any  other  region  N^.  which  is  con- 
nected with  S.  in  a  defniite  manner  suffers  at  the  same  time  a 
certain  definite  displacement  determined  by  the  displacement  of 
^';.  It  leaves  it  still  possible,  however,  that  the  dis])lacement  of 
N^.  may  depend  upon  the  manner  in  which  .'>'^.  is  connected  with 
S..  Take,  for  exam])le,  tiie  Euclidean  right  circular  cylinder,  and 
consider  two  strips  of  the  surface  connecting  the  same  two  points 
but  in  such  a  way  that  one  strip  winds  around  the  cylinder  more 
times  than  does  the  other.  The  same  motion  imparted  to  the 
same  end  of  each  strip  imj)arts  a  dilfei'cnt  motion  to  the  other 
ends. 

The  fifth  hypothesis  also  asserts  that  if  by  a  continu<Mis  dis- 
]daccment  N.  returns  to  its  original  jiosition,  so  does  also  N,.. 

7.  Thp:  Kxtended  ('(m^jdixate  Sv.STE^r. 
AVe  may  now  extend  our  cof'n'dinate  system  .r.  from  the  region 
7',  for  which  it  has  been  defined,  to  all  ])()ints  of  space.  For  that 
])urpose,  let  ns  consider  a  region  (jf  space  N,  com])osed  of  the 
])oints  whose  geodesic  distances  from  (J  are  less  than,  or  e(jual  to 
a  constant  7.',  where  /.'  is  less  than  the  smaller  of  the  two  (juan- 
tities  p  and  -HIj-,  p  being  the  length  of  the  shortest  geodesic  line 
which  can  be  drawn  iVom  O  in  7' and  JA  being  the  constant  men- 
tioned in  the  fourth  hvj)othesis.      Analytically  we  have  in  N, 


sin  /-.s' 
■'■,=",      /, 

,/•   =  cos  /:.< 


('=  h^,->) 


wliei'c 


a:.  =  1,        X    -_^  /.',       /!  .:_p,      I!  ■  ^    .,  • 


\\csh;ill  lii'.-t  ])l'u\'cthat  <r  n  i/ '/ri)il<yir  /  i  nr  en  ii  In  i  ikL  fi  h'iIi  1 1;  caii- 
tniii'il.  i'"oi' cnii-idci-  an\- geodesic  Hue  ''>0  in  N()l'h'ngth  //,  and 
lake  ( )^  ;i  poini  on  ' )<j  ^\\r\\  that  ' f( )^  =  ^  ■  -.  /.'.  There  exist.-  a  dis- 
pl;ieemeiit  -ueh  iliat  the  point   O  coi'i"e>|ioti(]s  to  ''>'|and  a  I'cgioii  7'^ 


FOKMS    OF    NOX-KrC'lJDKAX    SPACE.  Oo 

around  ()  c()n'os|)oiuls  to  a  reo'ioii  '/',  around  O^  in  such  a  manner 
that  the  ])ortion  of  the  geodesic  line  O^}  which  Hes  in  7',  corresponds 
to  the  ]>ortion  of  tlie  same  line  which  lies  in  1\  and  extends  in  the 
same  direction.  Here  7',  and  7',  are  both  contained  in  H^^,  but  by  vir- 
tue of  the  fifth  hypothesis  this  displacement  of  7j,  into  'i\  determines 
a  displacement  of  >'y  into  a  new  ])osition  S^.  The  line  OC^  of  lenij^th 
U  g'oes  then  into  a  line  0^(^^  of  the  same  length  ;  that  is,  the  line 
OQ^  has  the  length  /'  -f  L  Now  we  can  rejx^at  this  ojieration  with 
the  region  .^'^  by  selecting  on  O/^,  a  })oint  O.,  such  that  0^0,^  =  /, 
and  displacing  O^  into  0„  in  the  proper  manner.  In  this  way 
the  line  O^^  is  extended  indelinitely,  but  it  is  of  course  consistent 
with  the  theorem  that  the  line  should  be  a  closed  line. 

Axji  jtontl  in  .•<j)(U'e  nuiji  he  joined  lo  0  h\j  a  'jeexlem'  Vine.  A 
rigorous  j)roof  of  this  statement  may  be  given  by  means  of  the 
method  introduced  by  Hilbert  into  the  Calculus  of  Variations 
under  the  name  of  the  ''  J L'inftiiu/.'^veiifdJirenJ^'''  The  details  are 
too  involved  to  be  })resented  here.  \\'e  content  ourselves  with 
noticing  that  since  space  is  a  continuum  by  our  first  hypothesis, 
any  ])oint  P  may  be  connected  with  ()  by  a  continuous  curve. 
Now  the  Hilbert  method  consists  in  sliowing  that  among  all  the 
curv(>s  that  can  be  drawn  between  O  and  P  there  is  one  such  that 
no  other  has  a  greater  length,  and  that  tiiis  curve  in  sufficiently 
small  j)ortions  is  a  geodesic  line  as  we  have  defined  it. 

l>y  virtue  of  the  two  theorems  just  proved,  we  may  write 


sin  k-s 

('-l,-^ 

'■>) 

cos  hs, 

{"\  +  "l-^*A  = 

1). 

v.here  .s  is  unrestricted,  witli  the  assurance  that  all  values  of  x. 
thus  determined  represent  a  point  of  space  and  that  any  point  of 
space  may  be  re])resented  in  this  way.  This  is  our  generalized 
co()rdinate  system. 

fjct  us  take  now  any  point  P.      By  the  fourth   hypothesis,  the 

*  ('i)nsiilt  for  exninple  the  dissertation  of  ('has.  A.  Xoljle,  "  l-'.ine  nene  Metliode 
in  (ler  N'ariationsrecliuii.i^', "  (iottiiigen,  I'JOl. 


54  THE   BOSTON    COLLOQUIUM. 

region  S^  may  be  so  displaced  that  0  corresponds  with  i^  and  S^ 
with  a  congruent  region  xS  .  There  exist  then  relations  between 
the  coordinates  of  points  in  >S'^  and  the  coordinates  of  points  in  /S'^. 
We  shall  show  that  these  reldiionH  lutve  the  scane  form  c/.v  those 
ivhich  define  a  disphweinent  in  S^.  For  that  purpose  connect 
O  and  P  with  a  geodesic  line  and  take  on  this  line  the  points 
O,  (\,  (X,  ■■■  ,  (),=P,  such  that  the  distance  0.0.^^  is  less 
than  iu  If  then  O  is  displaced  so  as  to  coincide  in  succession 
with  Oj,  0^,  ■  ■  ■  ,  P,  there  is  determined  a  chain  of  congruent 
regions  S^^,  S^,  S.,,  ■  ■  •  ,  S^^,  eacii  of  which  has  points  in  common 
with  the  preceding  one.  The  displacement  of  ^S'^,  into  S^  however 
is  fully  determined  by  the  fact  that  a  region  around  0  is  dis- 
placed into  a  region  around  0^,  both  regions  lying  in  S^^.  Hence 
all  coordinates  of  all  the  jwints  in  S^  are  connected  witii  those  of 
/S'q  by  relations  of  the  form  given  in  paragra])h  5.  It  follows  that 
in  *S',  the  line  element  is  the  saine  as  in  aS',,,  that  a  linear  equation 
represents  a  geodesic  surface,  that  two  such  equations  represent  a 
geodesic  line,  and  that  a  displacement  of  a  portion  of  S^  is  repre- 
sented by  equations  of  the  same  form  as  in  .S'^.  In  like  manner 
we  can  proceed  from  -S',  to  S.„  and  hence  eventually  to  N  ,  tluis 
establishing  tlie  fact  to  be  })roved. 

It  is  clear  that  if  more  tiian  one  geodesic  line  can  be  drawn 
from  O  to  T*,  P  will  liave  more  than  one  set  of  coin'dinates  and 
more  than  one  s<'t  of  e<iuations  will  connect  the  cor)rdinates  of 
X   and  N,. 

Let  now  any  disj)la('ement  be  imparted  to  N,.  J3y  the  fifth 
hypothesis,  a  dis])lacenu'nt  is  then  imj)arted  to  X  through  the 
chain  N^,  N,,  N,,  .  .  .  ^  X  .  It  is  easy  to  see  that  the  analytic  ex- 
l)r('ssion  of  this  displacement  <)f  X'„  Avill  be  found  by  substituting 
in  the  displacenient  defined  for  N^  the  co<')rdinates  of  the  points  of 
X    determined  bv  the  chain  N,,  N,,  •  •  •  ,  N'  . 

We  may  now  establish  the  iiiq)ortant  j)roposition  :  //'/,•  /x  <i  recil 
(piif iihl I/,  frrri/  (/I  ixifsic  Imr  is  closed  diid  /la.s  a  leiK/th  not  e.rceedi ncf 
-■rrjkJ 

■  Tlii--  I  liciirfiii  is  (liif  til  1\  illiiiir.      His  ])r()(if  is  I'ssentialiy  tliat  tif  tlic  text. 


FOIIMS   OF   NOX-EUCLTDEAX    SIWCE.  •">•") 

For  j)ro()f  consider  a  point  Q  at  a  distance  Ttl'lk  from  0  on  the 
geodesic  line  .''.,=  <>,  x.^-^0.  The  cocu'dinates  of  (^)  are  (1  //,-,  0,  (\  <>). 
I^et  a  chain  of  congruent  regions  ^|„  N,,  N.„  •  •  •  ,  <9  ,  be  strung  ah)ng 
tlie  line  OQ,  tlie  ])oiiit  Q  lying  in  S'  ,,  and  each  region  being  ob- 
tained from  the  preceding  one  by  the  substitution 

sin  1:1 
■.t\  =  ./■,  cos  Li  +  ./;      j^     , 


a-',  =  —  xjz  sin  /c/  -|-  :r^  cos  Z7, 
where  I  <_  R. 

Apply  now  to  i>^^  the  displacement 

:r'^  =  .7'j  cos  0  —  :C2  -"^  4^> 
u:',,  =  o^j  sin  0  -f  .'■.,  cos  <^, 

3^9      =     ^.,1 


3" 


■   0' 


This  displacement  will  bo  transmitted  to  N^  through  the  chain 
8^„  /S'p  •  •  •,  /S\  The  distance  I)  between  the  new  and  the  original 
position  of  a  point  is  given  by 

cos  kD  =  xj'\^  +  Ir(:r\x[  +  a-^aC  +  .'■.^''3) 
=  xl  +  Jrj'l  +  F(aJ  +  ;>•:)  cos  (f) 
=  cos  (/)  +  (.'■;,  +  l:'x:)  (1  —  cos  (/)). 

Now  the  line  .^'i,  =  0,  .^-^  =  0,  a  ])ortion  of  which  lies  in  N  ,  is  dis- 

])laced   into  itself,  eacii  j)oint  being  moved   through  a  distance  J) 

wile  re 

cos  /•/)  =  cos  (f). 

Hence  as  </)  varies  from  0  to  27r,  tlie  point  Q  is  moved  on  .i\^  =  0, 
.r,  =  0  through  a  distance  'Itt  !:.  ]>ut  a  continuous  variation  of  (^ 
from  0  to  'Itt  restores  N^  and  hence  -s'^^  to  its  original  position. 
Hence  the  geodesic  line  x^^  —  0,  ,r^  =:  0  cannot  have  a  length  greater 
than  'Itt  /:.     The  theorem  is  thus  proved  for  a  particular  geodesic 


o6  THE    BOSTON    OOLI.OCH'ir.AI. 

lino;  but  !)y  ])ro})er  choice  of  the  origin  and  cointlinate  axes,  any 
geodesic  line  may  he  given  the  equations.*;,  =  0^  .v.,  =  0  and  lience 
I  he  theorem  holds  universally.  It  may  be  explicitly  noted  that 
we  have  not  })roved  that  a  geodesic  line  may  not  have  a  length 
less  than  'Itt'/:,  nor  that  all  geodesic  lines  have  the  same  length. 
A\'e  are  now  prejiared  to  prove  the  proposition  : 
To  (1/11/  Nc/  o/  r((li(('.-<  (.r^j,  .r^  ,r,„  .r.,)  saiisfi/inr/  tlic  Jandainenidl  rela- 
tion 

.rl  +  /-^(.r;  -f  .vl  +  x^)  =  1 

coiTfspnnd.s  one  (iiul  onhj  one  j)ohd  of  sjxicc. 

Tn  the  ])roof,  it  will  be  convenient   to  separate  the  three  cases 
of  zero,  negative,  and  ])ositive  curvature. 

1.  If/;  =  0,  the  coordinates  of  any  point  are 

,.,=  ar,     .r^^\.  (/=  1,2,3.) 

2.  If  /:  =  //;',  the  coinxlinates  are 

sinh  ///• 


'.].   If/'  is  real,  tlu;  coin-dinates  are 

sin  /■/• 
1  I      I.      ^        I 


=  cosh/-'y  (/=!,•_>,  3). 


•os/r  (/=  1,  :>,  3). 


It  is  now  I'eadilv  seen  that  if  the  quantities  .>\  are  given,  the 
(piantities  ",,  c.,,  "..,  r  are  unicjuely  determined  in  cases  1  and  2, 
e.\<'ept  for  sign  ;  while  in  case  3>  inultij)U\s  of  'lir  I:  mav  be  added 
to  /•  and  the  signs  ai'c  al>o  aml)iguous.  The  cliange  ol'siiin  of  all 
four  (piantities  (a^,  r)  does  not  alter  the  point  detei'mined  bv  them 
and  an  addition  of  "Jtt  /•  to  /•  in  case  3  amounts  simj)lv  to  travei's- 
ini;  the  h'ugth  ol' the  geodesic  line  one  or  more  times,  (iiven  the 
(piantities  ./•  therefoic,  we  lay  off  at  O  a  definite  direction  <i .  and 
measure  on  the  geodesic  line  with  this  direction  a  definite  dis- 
tance /■.      W  e  obtain  in  this  way  one  and  onlv  one  j)oint. 


FORMS   OF    NOX-KrOLIDFAX    SPACF.  -i  i 

S.    Tin:  Ai'xiLiAiiv  Space  1. 

The  (liscns.^ion  of  the  iollowinu:  j)aragrapli.s  will  l)e  clarified  hy 
iiiakinu'  ii>e  of  iainiliar  pi'dpo.sitions  of  the  ])rojective  o-eonietry. 
Ill  so  (IdiiiL;-,  we  avail  ourselves  of  theorems  which  are  in  essence 
analvtie.  Their  treometric  clothing  lends  vividness  to  their 
meaning  and  hel}>s  greatly  in  their  ajiplication.  We  consider 
then  a  projective  geometry  in  which  a  })oint  is  fixed  by  the  homo- 
U'eneons  coih'di nates 

^„  :  I.  :  I.  :  ^r 

A  lineal'  homogeneous  equation  defines  a  ])lane,  two  sut-h  ecjuations 
a  straight  line.  In  this  geometry  we  define  a  syston  of  projective 
measurc^ment,  hased  u])on  the  fundamental  quadric 

f:  +  iril;  -f  ^^  4-  ID  =  ^'- 

The  distance  A  between  two  points  is  by  definition  given   by  the 

relation 

Any  collineation  which  leaves  the  fundamental  quadric  invariant 
we  shall  call  a  movement  of  the  ])rojective  si)ace.  Such  a  move- 
ment leaves  distance  and  angle  unaltered.  The  space  in  which 
this  geometi'v  })revails  we  shall  call  the  auxiliary  space  2l. 

The  jKiints  ni'  S  may  l)e  made  to  correspond  to  the  points  of  S 
bv  ])lacintj:; 

where  the  sign  of  the  radical  is  the  same  for  all  values  of /.  It  is 
clear  that  geodt-sic  lines  and  surfaces  in  -S'  corres])ond  to  straight 
lines  and  ])lanes  in  ^  and  conversely.  Geodesic  distances  and 
angles  in  N  corres])ond  to  i)rojective  lengths  and  angles  in  ^  and 
a  disj)lacement  in  N  corresponds  to  a  movement  in  —  and  con- 
versely. 

Now  if  /•  is  zero  or  pure  imtiginaiy,  .';,  is  always  })ositive,  since 


^>X  Tin-:    I'.OSTON'    COLUKHirM. 

./•^  =  COS  /'.v.  IIciicc  ill  tlicsc  two  cases,  tlic  sii^n  of  the  radical  is 
uiianihigiioiis.  H  /:  is  real,  however,  .r^  may  be  either  ])()sitive  or 
neojative,  and  hence  either  sign  of  the  radical  may  !)C  taken. 
Hence  : 

If  k  is  zero  or  pure  iuKKjinary,  (i)ij/  point  of  "E  corrci^j^onds  to  one 
<ni(l  only  one  point  ofS;  vldle  ifk  is  real,  any  point  q/"^  may  cor- 
respond either  to  one  or  to  tiro  points  of  S  according  as  ,r.  and  — x. 
are  the  coordincifes  of  the  same  or  of  different  points  of  S. 

On  the  other  hand,  any  point  of  *S'  corresponds  to  as  many 
points  of  2  as  there  are  different  sets  of  coiirdinates  belonging  to 
the  })oint  of  N.  To  follow  this  more  in  d(!tail,  let  us  consider  the 
})oint  O  whicii  corresponds  in  S  to  the  point  (0:0:  0:1).  HO 
has  other  coordinates  it  must  be  possible  to  draw  a  geodesic  line 
from  (J  which  shall  again  return  to  0.  Tliis  follows  from  the 
ex])ressions  for  the  coordinates.  Let  us  call  this  line  r/.  C  orre- 
spondingly,  we  have  in  E  a  straight  line  y  connecting  two  points 
0  and  a',  each  of  which  corresjionds  to  ().  The  length  of  y,  and 
hence  of  7,  most  be  less  than  the  quantity  li  which  occurred  in 
the  definition  of  N, :  for  all  lines  of  length  /i*  or  less,  radiating 
from  O  determine  ))oints  in  S^^,  in  which  no  closed  line  is  ])os- 
sible.      Since  any  point  of  sj)ace  may  be  taken  for  0,  we  may  say  : 

Tiro  pointx  in  S  vdiieh  corrc.'^jiond  to  the  smne  jjoint  in  S  can  not 
he  iwarir  foi/efher  than  a  certain  Jiiiite  quantity. 

0.   1-'()i;ms  of  Si'ack  Which   Allow  Fi;kl  ^Io'I'iox  as  a 

W'jioLi;. 

We  are  to  examine  in  lliis  ])aragraph  the  I'esults  of  assuming 
that  the  di>|)laeenieiit  ol"  N  caused  by  a  disj)lacement  of  '^'^,  is 
iiide])en(lc!it  of  the  maiinei-  in  which  >'^^  is  coniUH'ted  with  N^ ;  that 
is,  it  i-  indepenilent  of  the  chain  of  bodies  X^,  X,  •  ■  •,  N.  In  this 
case  aiiv  displacement  of  N^  imparts  a  uiii(|iie  displacement  to  ("ach 
and  everv  ]»(iiiif  of  spa<'e.  W C  express  tliis  by  saying  that  space 
allows  free  motion  as  a  whole.      \\  e  assert  : 

//' .s'  tf/toirs  /jvv  iii'ifitiii  (IS  (I  irh(i/(,  <ini/  piiiiit  oj  S  cnrresponds  to 
iiiK    <iiid  iiidii  din    poitil  (i[   i. 


Fomrs    OF    NOX-FrCLIDlOAX    SPACE.  o9 

Consider  :i  point  /*  in  >',  and  let  lis  assninc  that  /'  corresponds 
to  two  points  11  and  11'.  As  sliown  in  tlH>  last  pai'agraj)!),  if  II 
and  IT  are  coinieeted  l)v  a  straiixht  line  7,  there  will  correspond  in 
N  a  line  y  which  starts  from  P  and  I'etnrns  to  the  same  ])oint. 
Along;  this  line  we  may  constrnct  a  chain  of  contrnient  regions  H^, 
N,,'S'.„  ■  ■•,  >',|,  where  S^^  is  the  same  region  as  ."^'i,.  Corresponding; 
to  this  contigin'ation,  we  have  in  S  a  chain  of  regions  3^,,  ]^j,  S.„ 
•  ••,  S  ,,  where  S  is  distinct  from  S^^.  Xow  any  displacement  im- 
parted to  >^^  is  transmitted  throngh  the  chain  6'^,  aS',,  •  ■  •,  N_  back 
to  ^^^.  Bnt  this  displacement  of  ><^^  mnst  be  the  same  as  that  of 
Nj,  if  sj)ace  is  movable  as  a  whole.  If,  f)r  exam})le,  N,  is  so  moved 
that  all  points  on  a  geodesic  line  /  are  fixed,  S^^  must  be  moved  in 
the  same  manner.  Corres])ondingly,  we  must  have  in  S  a  dis- 
placement by  which  two  straight  lines  X  and  X',  one  lying  in  S,j, 
the  other  lying  in  2^^  ai"e  each  point  for  point  fixed.  This,  how- 
ever, is  impossible  unless  2£^  coincides  with  S,^.  Hence  the  as- 
sumption that  y-*  corresponds  to  two  points  H  and  IT'  is  untenable. 

SjMices  of  Zero    (_'urr((ture. 

If  k  =  0,  the  relation  between  points  of  »S'  and  those  of  —  is  one 
to  one.  In  other  words,  to  each  ])oint  of  N  corresponds  one  and 
onlv  one  set  of  coordinates  x.  and  conversely.  We  have  there- 
fore a  geometry  in  which  the  theorems  of  paragraph  o  hold  univer- 
sally. In  addition  all  geodesic  lines  are  infinite  in  length.  We 
may  consecjuently  introduce  the  conception  of  jiarallel  lines  l)y  the 
following  definition  :  A  line  A  B  is  j)arallel  to  CJ)  when  AB  is 
the  limit  approached  by  a  line  .IC'  intersecting  CD,  as  the  point 
of  intersection  recedes  indetinitely.  It  may  then  l)e  shown  that 
through  any  ]M)int  of  space  there  goes  one  and  only  one  geodesic 
line  which  is  })araliel  to  a  given  geodesic  line  not  passing  through 
the  given  point.     The  resulting  geometry  is  the  Kncl'uhai)  < i<  inmtrii. 

Spdres  of  CoiiKfdiit  Xcf/djirc  (/lUTittu re. 

If/;  is  pure  imaginary,  again  the  relation  between  the  jtoiiits  of 
N  and  those  of  2  is  one  to  one.      A\'e  have  again  a  sj)a(,e  in  which 


<)0  TKE    BOSTOX   COLLOQUIUM. 

the  tliooronis  of  paragraph  5  hokl  universally  and  in  which  all 
o-eiHlesie  lines  are  inlinitc  in  length.  If  parallel  g(!odesic  lines  are 
defined  as  for  /;  =  0,  then  through  a  given  point  there  go  two  and 
only  two  geodesic  lines  parallel  to  a  given  geodesic  line  not  pass- 
ing through  the  given  point.  All  other  geodesic  lines  through 
the  point  and  lying  on  the  geodesic  surface  determined  hy  the 
given  ]ioint  and  the  given  geodesic  line  are  sei)arated  by  the 
l)arallel  lines  into  two  classes,  consisting  respectively  of  the  lines 
which  do,  and  of  the  lines  wiiich  do  not,  intersect  the  given  geodesic 
line.      The  geometry  is  the  Lohdcheri^kian  flcouuirji. 

SiJ'ioc!^  (if  ('()ii>;tant  Positive  ('nrrcfi'rc. 

Tf /•  is  real,  two  cases  present  themselves.  In  the  first  case,  the 
relation  between  the  points  of  S  and  those  of  i  is  two-to-one. 
Then  to  each  point  of  S  corresponds  only  one  set  of  co(>rdinates 
and  conversely.  In  particular,  the  co()rdinates  .r.  and  —  ,'■.  belong 
to  different  jwints  of  sjiace.  The  theorems  of  })aragra]>h  5  hold 
only  in  a  restricted  ))ortion  of  s|)ace  in  which  the  greatest  geodesic 
distance  is  tt/Zc.  All  geodesic  lines  are  closed  and  of  length  equal 
to  27r//.-.  Two  intersecting  geodesic  lines  intersect  again  at  a  dis- 
tance Trjf:  on  each  of  them  from  the  first  point  of  intersection, 
"^riiere  are  no  parallel  lines  in  the  sense  of  the  definition  given  for 
/•  =  0.  In  fact  any  two  geodesic  lines  on  the  same  geodesic  surface 
intersect.  All  geodesic  lines  ])erpendicular  to  the  same  geodesic 
surface  intersect  in  two  points  which  are  distant  ttJ'II:  from  the  sur- 
face.    The  geometry  is  that  called  by  Klein  {\\v  Sp/u'ricfif  (iennuiry. 

In  the  second  case,  the  relation  between  the  points  of  N  and  those 
of  ]!l  is  one-to-one,  in  the  sense;  that  to  each  point  of  N  belongs  the 
two  sets  of  coln'dinates  .i\  and  —  ,'• .  The  theorems  of  ])aragra]ih 
■")  hold  for  a  poi'tion  of  space  in  which  the  greatest  geodesic  dis- 
tance is  rrjl:.  All  geodesic  liues  ai'e  closed  and  of  a  length  tt j k 
and  anv  two  iiitei'secting  geodesic  lines  return  to  tlie  point  of  in- 
tei'-ectioii  without  j)i'eviou--l v  lueeting.  All  geodesic  liues  per- 
pcndiciilai'  to  tiie  >ame  getidesic  surface  meet  in  a  point  at  a  dis- 
t.iuce  tt/l'/,  from  the  surface.  The  geometrv  is  called  by  Klein 
the    l-'JIijiln-   <  <<(iiii(irii. 


FOlOrS   OF    NOX   FrcLII)i:AN    SPACF.  Gl 

We  may  sum  nj)  as  follows  : 

The  nnlt/  .^jxtccs  sadsfi/inf/  our  fire  /ii/jxidic'ics  and  (ddyviiuj  free 
iiiodoii  a.s  <t  vhole  are  die  J-jicddedit,  IjohncherHhian,  Sjtlui'ical  cikI 
J^//ij)f!c  ■•<j>(ices. 

10.    FoK.Ms  OK  Si'ACK  Which  do  Not  Allow  Fi;ke 

]\r()TI()N    AS    A    WjIOLK. 

V\e  consider  next  spac(>s  in  wiiich  the  displacement  of  N^  caused 
by  the  displacement  of  >S ,  is  dependent  upt)n  the  manner  in  which 
'S_  is  connected  with  N,.  These  are  called  by  Killing  the  Clitford- 
Klein  s})ace.      They  have  been  illustrated  in  parag'ra})h  G. 

From  what  has  ])recedcd,  it  is  clear  that  in  the  Clifford-Klein 
spaces  a  ])oint  must  have  more  than  one  set  of  ,';^-co(')rdi nates. 
Consider  then  the  region  N^  and  let  •'',■  be  one  set  of  coordinates  of 
its  points.  Then  \i\r'.  are  also  the  coiu'dinates  of  its  jioints,  :r'.  may 
be  obtained  from  .'■.,  as  we  have  seen,  by  following  out  a  chain  of 
disjilacements  by  which  ^',  takes  in  succession  the  ])ositions  N,,  S^^ 
,S'.„  •  ■  •  N  =  N^.  That  is  x'.  and  .r.  ai'e  connected  by  relations 
which  have  the  form  of  the  displacement  formulas.  Suj)pose  these 
relations  denoted  l)y  7>,.  Let  now  i/.  be  the  cooi'dinates  of  a  point 
P  lying  outside  of  N,.  It  may  be  connected  with  S^  by  a  geodesic 
line  and  a  chain  of  regions  >'|,,  N,,  ■^'.,,  ■  ■  ■  ,  >''  constructed  along  this 
line.  If  the  disjilacement  />,  is  im])osed  upon  >',|,  it  will  be  trans- 
mitted to  >''  ;  and  since  S^^  returns  to  its  original  ])osition  the  same 
is  true  i>l"  ■'^',',  by  the  fifth  hypothesis.  That  is  the  transformation 
/>,  givt's  a  relation  between  two  sets  of  coiirdinates  of  any  ])oint  of 
space.  Such  a  transibrmntion  is  said  by  Killing  to  re])resent  the 
coincidence  of  jxtints. 

Jt  is  clear  that  the  inverse  transformation  7>|  '  also  rej)i'e>e]its 
the  coincidence  of  jxtints,  and  ii"  1)^  and  />.,  each  repi'csents  the 
coincidence  of  points,  the  translbi'ination  I\P.,  does  also,  and  this 
is  ti'U('  when  7>.,  is  the  same  as  />,.  That  is,  f/ie  fi'(iitsjniuiirfd()))s 
irJt/cli  rrprrxrni  flic  cot iicidcnee  of  poiiitx  nt  sjKtce  Jonn  <t  (ji'oKp. 
This  gr(Uip  we  shall  call  the  ;/roi(p  of  die  ^jxtce. 

The  grou])  of  the  space  inter])ret(>d   in  !l  is  a  group  of  collinea- 


i')-2  THE    BOSTOX    COLLOQl^I^M. 

tioiis  l)v  which  the  fuiKlamental  quadric  is  invariant  and  hy  which 
points  that  correspond  to  tlic  same  point  in  S  are  transformed  into 
each  other.  J>ecause  of  the  theorems  established  in  paragraph  8 
it  follows  that  the  group  of  the  s})ace  inter})reted  in  2  must  not 
onlv  he  properly  discontinuous  but  must  be  subject  to  the  condition 
that  the  distance  between  corresponding  points  shall  never  be  less 
than  a  certain  finite  quantity.  In  particular,  no  fr(ii)s:formfifion 
of  f/ic  'jroiip  111(11/  Jiarc  a  real  jixed  point.  If  the  region  of  discon- 
tinuity of  the  group  in  w  is  obtained,  this  region  will  correspond 
in  a  one-to-one  manner  to  S',  when  /.■  is  zero  or  pure  imaginary, 
and  in  either  a  one-to-one  manner  or  a  one-to-two  manner  to  *S' 
when  I:  is  real.  Conversely,  the  region  of  discontinuity  of  any 
properly  discontinuous  group  in  2,  l)y  whicii  the  distance  between 
two  corresponding  points  is  never  less  than  a  finite  (juantity,  will 
furnisli  an  example  of  a  space  satisfying  the  five  hypotheses, 
rience  the  ]>roblem  to  determine  the  Clifford- Klein  space  is 
reduced  to  the  })roblem  to  determine  all  groups  with  the  re(piired 
j)ro})erties. 

IJefore  j)roceeding  to  the  nearer  discussion  of  the  problem,  we 
mav  note  that  our  derivation  of  the  group  of  the  space  is  based 
upon  the  consideration  of  a  three-dimensional  region  S^  in  which 
each  point  has  difi'erent  sets  of  cor)rdinates.  This  region  gives 
opportunity  to  a])i)ly  the  fifth  hyj)otliesis.  There  is  still  the  i)os- 
sil)ilitv  thci'efore  that  certain  excejjtional  one-dimensional  or  two- 
diincnsioiial  regions  may  exist,  upon  which  the  same  point  may 
have  ^<'t<  of  cix'M'dinates  not  coimected  by  transll)rmations  of  the 
group.  The  following  two  exain])les  are  given  by  Killing  of  a 
two-diiiirii<ional  sj)acc  of  zero  curvature  having  an  i'Xcej)tional 
line. 

1.  (  "onsider  a  cvliiider  in  I-]uclidian  >j)ace  standing  uj)ou  a  cubic 
cur'sc  with  a  double  point.  The  gconietrv  of  tlie  cylinder  is  that 
ofth-'   I'.ncliilcau  j)lanc  except   for  the  presence  oi"  the  doul)le  line. 

We  e;ill  'Id  the  length  of  the  loup  of  the  cubic,  and  take  as  the 
oi'iu'in  of  corii'dinates  tiie  point  on  the  loup  e(|nitlistant  from  the 
double  point  ill  each  dii'ection.      Then  ii'  we  tais'e  for  one  eoi'ii-dinate 


FORMS  OF  Nox-i:rcL[i)i:AX  sPAci:.  o;> 

the  k'liixth  .s  of  the  cubic  and  for  the  other  the  length  li  of 
an  element  of  the  cylinder,  the  coordinates  (.v,  A)  correspond  in 
a  one-to-one  manner  to  the  points  of  the  surfac^e,  except  that 
tile  coordinates  [d,  h)  and  (—  (i,  h)  correspond  to  the  same  point 
of  the  surface. 

■J.  ( 'onsider  a  cylinder  in  iMiclidean  space  standing*  on  a  lemnis- 
cate.  Its  geometry  is  the  same  as  that  of  the  Euclidean  plane 
for  restricted  portions.  We  will  take  the  origin  at  the  double 
point  of  tlie  lemniscate,  define  s  as  the  length  of  the  curve  and  h 
as  the  length  of  an  element  of  the  cylinder.  Then  if  2a  is  the 
entire  length  of  the  lemniscate,  the  group  of  the  surface  is 

.S-'  =  .s  -)-  2n((, 
h'  =  h, 

where  n  is  an  integer;  that  is,  the  coordinates  (.s,  h)  and  {s  -f  liui,  h) 
refer  to  the  same  point  of  the  surface.  But  the  coordinates  (0,  h) 
and  {n((,  li)  also  refer  to  the  same  point  of  the  surface,  since  they 
give  points  on  the  double  line. 

Examples  of  a  similar  kind  may  be  formed  for  three  dimensional 
s])aces  without  difliculty  as  far  as  the  analytic  work  is  concerned. 
J  low  far  they  are  conceivai)le  as  an  explanation  of  ])hysical  space, 
involving  as  they  do  the  passing  of  space  through  itself  without 
break  in  tiie  continuity  of  each  of  the  intersecting  portions  may 
be  open  to  question.  They  have  been  examined  by  no  one  in 
detail  and  we  shall  rule  them  out  of  the  following  discussion. 

We  pass  now  to  the  special  consideration  of  the  three  kinds  of 
space. 

Spdccs   of  Zero    ('nrrafid'e. 

If  /;  =  0,  S  is  the  Euclidean  space  and  its  movements  are  the 
Euclidean  movements.  A  rotation  around  an  axis  cannot  be  a 
transfn-mation  of  the  grouj)  of  the  sjiace  N  since,  as  we  have  seen, 
no  transformation  of  tlie  group  can  have  a  real  lixed  j)oint.  We 
must  form  the  group  therefore  by  the  use  of  translations  and  screw 
motions. 

The    use   of   translations   alone    lead  to    three   and    onlv   thi'ce 


04  Tin-:  BOSTON  coLLOQriu^r. 

])roperly  diseoiitinuous  groups,  having  for  regions  of  discontinuity 
ros])e('tivcly  : 

[(()  A  j)arallelopipotl  with  three  finite  edges. 

(/>)  The  limiting  figure  of  a  ])aralleloj)iped  when  one  edge  be- 
comes infinite. 

(c)  The  limiting  figure  of  a  ]iarallelopi})ed  when  two  edges  be- 
come infinite. 

The  geometry  in  .S  may  be  readily  constructed  bv  oj)erating 
with  the  Euclidean  geometry  in  the  regions  (a),  (/>).  (r),  resjH'C- 
tivelv.  \\'henever  a  straight  line  meets  a  bounding  face  of  the 
region,  it  is  continued  from  the  cori'esj)onding  point  of  the  op))0- 
site  face.  For  brevity  we  shall  mention  without  ])roof  some  of 
the  results  in  case  (a). 

Some  geodesic  lines  are  closed  and  some  ar(>  infinite  in  length 
and  those  which  are  closed  are  not  all  of  the  same  Icngtli.  In 
fact  geodesic  lines  can  l)e  di'awn,  liaving  the  finite  length 
/a  +  iiih  4-  ;/(',  where  r/,  A,  c,  arc  the  lengths  of  the  edges  of  the 
parallel(»piped  and  /,  ///,  n,  ai'c  any  three  I'clativcly  pi'ime  integers, 
(rcodesic  surfaces  are  of  three  l<inds.  Some  are  indefinite  in  ex- 
tent, possessing  no  points  with  moi'e  than  one  set  of  co("\rdinates. 
On  th(\-e  tile  gcomcti'v  is  identical  with  the  JMiclidean  ge()metry. 
Others  are  I'eprescnted  in  !^  bv  a  strip  of  a  ])lane  bounded  by 
j)ai'allel  Hues  and  have  in  N  the  connectivity  and  geometrv  of  a 
I'Jiclidean  evliiidei".  Othei-  surfaces  arc  rcpi'csented  in  ^  by  a 
])laiie  parallelogram  and  have  in  .^'  the  coiuiectivitv  of  a  ring 
sui'facc. 

Xo  exhaustive  studv  has  \)ovn  made  of  the  ( 'lilford-K lein  spaces 
\vho-c  groups  contain  screw  motions.  In  fact  l\lein  sa\s,  without 
])roof,  that  a  sci'cw  motion  i-  not  allowable,  but  Killing  gives  the 
following  two  examples  which  seem  valid  : 

(")   TIh'   Li-i'oup   of   the    s|)ace    is    genci'atc(l    bv   a    <iiiule    sci'cw 

mot  ion  : 

.'•'  =  ,''|  co^  7.  —  ,/■,  sin   7, 

.'■',  =  .'',  -in   7  4-  ,c,  cn<  7, 

.'■;  =  .'•,  -f  /', 


FOiniS   OF    NOX-KUCLIDKAN    SPACE.  Go 

wliere  i  and  A  arc  constants.  The  reo;i<)n  of  discontinuity  in  S  is 
then  bounded  by  two  parallel  planes  at  a  distance  //  units  from 
each  other. 

(A)  The  (^roup  of  the  sj)ace  is 

:>•:  =  (_l)'.r,  +  nh, 

•*";i  =  -''a  +  ^''^ 
where  a,  h^  c,  arc  constants  and  /,  //?,  /(,  are  arbitrary  parameters. 

We  i^ive  without  proof  some  of  the  striking;  peculiarities  of  *S'  in 
the  case  {(i). 

There  is  a  unique  geodesic  line  of  length  //  which  we  shall  call 
the  axis  of  the  space.  If  a  and  ir  are  incommensurable,  this  is 
the  only  closed  geodesic  line  ;  if  a  and  tt  are  commensurable,  all 
geodesic  lines  parallel  to  the  axis  are  closed  and  of  lengths  equal 
to  multiples  of  //,  J'^or  all  values  of  a  there  are  geodesic  lines 
with  double  points.  Through  any  point  of  space  there  goes  an 
infinite  innnber  of  such  geodesic  lines  liaving  the  given  point  for 
a  doul)le  point;  and  for  a  given  direction,  not  parallel  or  perjien- 
dicular  to  the  axis,  there  exist  an  infinite  number  of  geodesic  lines 
with  double  points.  Geodesic  surfaces  are  of  three  kinds  :  (1) 
those  ])er]KMidicular  to  the  axis,  (2)  those  jiarallel  to  or  contain- 
ing the  axis,  (•'])  those  which  have  neither  of  these  relations  to  the 
axis.  On  geodesic  surfaces  of  the  first  kind,  all  geodesic  lines  are 
infinite  in  length  and  the  geometry  is  that  of  the  Euclidean  plane. 
On  geodesic  surfaces  of  the  second  kind,  there  are  no  closed 
geodesic  lines  but  a  geodesic  line  may  have  a  double  })oint.  On 
geodesic  surfaces  of  the  third  kind,  all  kinds  of  geodesic  lines  lie. 
The  last  two  kinds  of  surfaces  present  the  peculiarities  of  cylinders 
with  double  lines  mentioned  on  pp.  ()"2— ']. 

Sp(((U's  of  (hiisfdiif  l'(>.-<ifirc  (Uirvaturv. 
If/,-  is  I'eal,  thei'c  is  u  fundamental  difference  between  sjiaces  of 
an    even  and   those  of  an    odd   numlx'r  of  dimensions.      It  is  a 
simple  matter  to  apply  our  foregoing  discussion   to  s])ace  o'i   two 


6()  TIIK    P.OSTOX    ('( )!.].( X^r I TM. 

(limeiisions  by  (li'()])i)iiig'  the  eix'n'dinate  .r.j  and  niakiiicj:  neee.ssarv 
c'lianii^es.  It  ai)i)eai's  tliat  any  (lis])lac'eiiieiit  has  a  real  fixed  point 
and  c'onsecjiiently  there  can  be  no  gronp  oi' the  space.  If  we  rnle 
out  such  special  lines  and  points  as  occur  on  cylinders  with 
double  lines,  we  are  then  led  to  the  discussion  of  i)aragraph  9. 
Hence  the  theorc^n  : 

.1  iioii-ciic/idcdii  .■<j}(i('c  of  tiro  (Jiine)ifiioii.s  and  of  conatdnt  positive 
cuiTdfiirc  for  v/iic/i  oar  hj/potheses  liold  (iiid  i)i  jrhich  uo  special 
points  c.ri.sf  lios  the  co iinccticitij  (ind  fJte  f/eovwtri/  (f  either  tlie 
sphericol  .^iirj'dce  or  the  ellijAic  plane. 

Consider  now  a  space  of  three  dimensions.  The  study  of  the 
collineations  which  leave  invariant  the  qnadric 

i;:  4-  ir{^  +  e,  +  id  =  ^\ 

and  which  we  call  the  movements  of  !l,  lead  to  the  following 
results. '■'  l)y  any  real  movement  in  ^  two  i-eal  lines  G  and  11, 
recij)rocal  polars  with  resj)ect  to  the  fundamental  ([uadric,  are 
unaltered  as  a  whole,  each  ])oint  on  each  of  the  lines  being  dis- 
])laccd  through  a  distance  which  is  constant  for  that  line.  It"  the 
<lis})lacemcnt  is  diiferent  for  the  two  lines  (j  and  7/,  these  are  the 
only  fixed  lines.  If  however  the  displacement  is  the  same  for  (i 
and  JI,  then  all  lines  of  a  certain  line  congruence  arc  fixed,  this 
congruence  being  made  \\\)  of  all  lines  which  intersect  the  same 
two  conjugate  imaginary  generators  of"  the  f"undamental  (piadric. 
Anv  j)oint  of  ^  is  then  displaced  a  constant  distance  along  tiie 
line  of  the  congruence  which  contains  the  j)oint. 

Such  a  transformation  is  the  nearest  analogy  in  a  space  of  con- 
stant ])ositive  curvature  to  a  translation  in  iMiclidcau  s])ace.  It 
is  accordingly  caih'd  a  frdnsldfimi,  and  the  congruence  of  fixed 
line-  arc  <'ancd  ('lifi'ord  pordllcls.  The  name  |)ai'allels  is  sug- 
gested l)\-  the  I'clation  of  these  lines  to  a  translation,  but  thev 
ha\('  othei'  j)ropcrties  analogous  to  those  of  the  lMicli<lean  par- 
allel-^.      I*'or   example,  iVom    any    point    in   I'itlicr   of  two   Cliflbrd 

('musuIi    tnr   tlh'  (Irtails  (if   the   L;('iimcti-_v   nl'   this   [iar:iL,'r;iiili  :     l\loiii,    "Ziir 
Niclil-luikll.lisrli.-ii  <.«■(. iiictric,''   .1/'////.  .!///(. 'A//,  vol.  :;7  (IM»l)),  y.  :,  I  1. 


FOIOrS    OF    XOX-EUCIJDKAX    SPACE.  G7 

parallels  u  coniiiion  perpendicular  can  be  drawn  to  the  two,  and 
the  portion  of  the  [)erpen(li('ular  included  between  the  two  has 
always  the  -anie  length.  Again,  if  a  line  cut  two  Clifford  par- 
allels the  corres})onding  angles  are  equal. 

The  Clifford  parallels  are  of  two  kinds,  according  as  the  gener- 
ators of  tlie  fundamental  quadric  which  determine  them  l)elong  to 
one  or  the  other  of  the  two  sets  of  generators  of  the  quadric. 
Similarly  we  must  distinguish  between  two  kinds  of  translations. 
Two  translations  of  the  same  kind  carried  out  in  succession  are 
equivalent  to  a  translation  of  the  same  kind,  but  two  translations 
of  different  kinds  are  not  equivalent  to  a  translation.  Hence  the 
translations  of  each  kind  form  by  themselves  a  group. 

Let  us  consider  first  non-euclidean  spaces  whose  groups  are 
formed  by  translations  alone.  These  translations  must  all  be  of 
the  same  kind.  If  we  place  1:  =  1,  for  convenience,  and  intro- 
duce \  and  iJL  as  the  parameters  of  a  point  on  the  fundamental 
(piadric,  whereby 

,__^. +  /!.._     I3-/I0 

then  any  translation  of  the  one  kind  causes  a  substitution  of  the 
form 

(</  -f  /c)\  -  {h  -  ia) 
^    ~  {h  +  hl)\  -f  {<1  -  ic)  '  ^    -  ^ 

and   conversely. 

On  the  other  hand,  if  we  interpret  X  in  the  usual  manner  as  a 
complex  variable  ujwn  the  vniit  sphere,  the  above  substitution 
represents  a  rotation  of  the  sj)liere.  To  any  translation  of  the  one 
kind  in  2  corres))onds  then  a  rotation  of  the  sphere,  and  in  fact 
the  angle  of  rotation  of  the  sphere  is  ecpial  to  the  distance  by 
which  the  ])oints  of  S  are  displaced  along  a  system  of  Clifford  par- 
allels. The  group  of  the  space  corresponds  to  a  group  of  rotations 
of  the  sphere,  and  since  the  amount  of  displacement  by  any  trans- 
formation of  the  group  is  never  less   than  a  finite  quantity  B  it 


C8  TJIE    BOSTON    COLTXXH  lU.AI. 

follows  that  the  ^rou])  of  rotations  can  contain  no  iiitinitesimal 
rotation.  This  condition  is  met  only  by  the  groups  of  rotations 
by  which  a  regular  ])olyedron  concentric  with  the  sphere  is  trans- 
formed into  itself.      We  have  accordingly  the  theorem  :* 

//'  (I  Clifford-Kh'hi  space  of  comtdnt  positive  curvature  is  frans- 
forined  into  itself  by  a  (/roup  of  tr((iistations,  this  ejroup  must  be 
holoedric-isomorph  vith  a  group  of  the  ref/ular  polt/edi-a  ;  and  con- 
versely, to  any  group  of  the  regular  polyedra  cor  res j)ond  four  spaces 
of  constant  positive  curvatnre,  according  as  tJie  coordinates  x.  and 
—  X.  represent  the  same  or  different  points  of  space  and  as  the  group 
of  the  space  is  made  up  of  transhdions  of  one  or  the  other  kind. 

It  remains  to  ask  if  groups  of  the  space  may  contain  displace- 
ments which  are  not  translations.  This  question  is  answered  in 
the  negative  by  Killing  (/.  c.)  but  his  proof  is  not  satisfactory.  He 
shows  conclusively  that  if  I)  is  a  dis])lacement  belonging  to  the 
group  of  the  s})ace  and  if  (i  and  //  are  the  two  fixed  lines,  then 
the  smallest  displacement  along  either  line  caused  by  the  re])eti- 
tion  of  I)  must  be  77/7,  where  (j  is  an  integer,  the  same  for  both 
lines.  Rut  he  errs  in  assuming  that  this  minimum  displacement 
is  caused  in  both  lines  by  the  same  transformation.  For  exam})le, 
consider  the  dis])]acem(nit  I) 


IT 

.       TT 

'■'  '••*^  :. 

—  ■ 

'■.,  sm  .  , 

.    7^ 

vr 

=  ./ 

.^'"5 

+  ■ 

.■iTT 

.     •>7r 

=  J 

'  .  C(  )S 

—  J 

"  sm    _ 

and    the   group  />,   />-.   //'.   />',   //' =  1.      The  two  fixed    lines  are 

■  'I'hi-  llicorcni  !■-  new  ri-  f:ii' ;i<  t  lie  :iul  lim-  know-.  l\illii),u  {  ( .' m  ii>ll,ii;,  ti  <h-r 
C'.iiii  irn  ,  viil.  1,  p.  ,">  11  )  iKitiies  tli;il  if  tiic  Lri-inili  nf  ;i  spiicc  ot'  /.  1  cimi;!!!!-;  ;i 
tiMii-latioii,  tlic  ;iin(iunt  tA'  tin'  t  riiiisl.ition  must  l)f  ;m  :ilii|n(it  part  of  -,  Imt  he 
Icnvc-  the  iiii|iri'<sinii  that  :m_v  thfcr  --ucli  traiishit  icii-  may  he  cnmliiiicil  at  jilcas- 
wvc  tci  form  a  ;_'rni:p  uf  a  sjiaci". 


FOinrS    OF    NOX-ErCLTDKAX    SPACE.  69 

r;  (.Cj  =  0,  .v.,  =  0)  and  7/'(.r,j  =  0, .);,  =  0)  and  the  smallest  di.<i)lace- 
nient  along  each  is  tt/o,  J>ut  this  di.sj)laceineiit  is  j)r()duced  along 
(r  by  J)  and  J)*  and  is  produced  along  J[  hy  7/"  and  //'.  ]>y  no 
substitution  of  the  group,  however,  can  the  distance  between  two 
corresponding  points  fall  below  a  definite  finite  quantity.  Hence 
the  group,  which  is  not  composed  of  translations,  is  allowable  as 
the  group  of  a  non-euelideau  space.  The  investigation  of  such 
groups  is  yet  to  be  made. 

C'///fb/v/',y  Sti.rf"-Ce  of  Zero  Ciirvafjtre, 

It  is  of  interest  at  this  point  to  mention  Clifford's  surface  of 
zero  curvature  and  finite  extent  which  first  led  to  the  conception 
of  the  Clifford-Klein  spaces.  This  surface  may  be  obtained  by 
choosing  on  the  fundamental  quadric  of  the  above  space  of  con- 
stant positive  curvature  two  conjugate  imaginary  lines  from  each 
set  of  generators.  The  quadric  surface  which  passes  through  the 
qn.adrilateral  thus  formed  is  the  surface  required.  It  is  clear  that  the 
surface  contains  two  sets  of  Clifford  parallels  and  is  transformed 
into  itself  by  two  translations.  If  we  take  the  four  lines  on 
the  fundamental  quadric  as  corres]:)onding  respectively  to  A.  :=  0, 
A,  —  oc,  /Lt  =  0,  and  /x  =  x  in  our  previous  notation,  the  equation 
of  the  surface  is 

IJ  +  I^  -.r(f^  +  |^)  =  0, 

where  ({  is  a  real  constant. 

\Ve  may  define  the  two  sets  of  Clifford  parallels  on  the  surface 
by  the  })arameters  n  and  r,  where 

_  ^1  +  "^?,  ^       I,  -  <'^.  ^  ^^ 

To  obtain  the  line-element  of  the  surface,  we  write  first  ^.  =  px., 
where  p'  =  ^^;'.      Then  for  the  space 


70  THE   BOSTON   COLLOQUIUM. 

and  this  applied  to  the  surface  gives 

du'''  a"^  —  1  dadv  dv^ 

'  "'  =  (1  +  irf  ~  -  a'  +  1  (1  +  w^)(l  Tv')  '^  {l~+vf ' 

If  we  take  next  as  parameters  a  and  t  the  lengths  of  the  gener- 
ators, by  placing 

r  du  r   dv 

the  line  element  takes  the  simpler  form 

«'  -  1 

d,r  =  da-  —  2    .,         -  da-dr  +  dr^. 

a^  -f  1 

From  this  it  appears  that  the  Gaussian  curvature  of  the  surface 
is  zero. 

The  relation  between  a  point  of  the  surface  and  a  value-pair 
((/,  r)  is  one-to-one.  Hence  if  r  is  kept  constant  and  it  varies  from 
—  oc  to  -|-  oc  the  corresponding  i)oint  describes  a  generator  once. 
At  the  same  time  a  varies  continuously  from  —  tt  2  to  tt  2.  The 
total  length  of  a  generator  is  then  tt,  a  finite  quantity. 

The  area  of  a  })ortion  of  a  surface  of  which  the  line  element  is 

d.s-  =  Edir  -f  -IFdudv  -f  Gdv^ 
is  defined  bv  the  doul)le  intetrral 


// 


1   EG  —  F-  dude 


taken  over  the  j)»)rtion.      Hence  the  total  area  of  the  Cliilbrd  sur- 
face is 


iladr  =     „ 

,t-  +   1 


W'c  have  tlicfcforc  an  exatnple  of  an  inibounded  surface  of  zero 
curvatui-c  iijxni    which    the  coiuicctivitv  and    the  geometry   is   that 


FORMS   OF    XOX-EUCLIDFAX    SPACE.  71 

of  a  parallelogram  on  the  Euclidean  plane,  the  opposite  sides  of 
the  jiarallelograni  corresponding  point  for  point. 

Spaces  of  Constant   Xegative   ('urvature. 

Yi'  I:  is  pure  imaginary,  we  may  place  k  =  i  for  convenience. 
AVe  have  then  in  2  the  fundamental  quadric 

^]  +  e  +  ^i-^i  =  0, 

points  in  tiie  interior  of  which  correspond  to  real  values  of  .v.  and 
hence  to  real  points  of  N.  Any  collineation  wliich  leaves  this 
quadric  invariant  determines  a  linear  substitution  of  the  param- 
eters \  and  \  where 

,    i  +  ii      .    i,-ii, 

and  conversely  any  })air  of  linear  substitutions 

where  the  determinants  aS  —  /3y  and  aS^— /^yare  not  zero,  de- 
termines such  a  collineation.*  These  collineations  are  the  move- 
ments of  S.  A  real  movement  occurs  when  and  only  when 
a,  /3,  7  and  ^,  are  conjugate  imaginary  to  0;,  /3,  7,  8,  respect- 
ively.     A  real  movement  may  consequently  be  determined  by  the 

single  substitution 

,       aX  +  /3 
^   ==  7\  +  §  • 

Let  us  sui)))()se  first  that  the  substitution  in  X  leaves  two  dis- 
tinct values  of  X  unaltered.  There  correspond  two  fixed  points 
on  tlie  fundamental  (juadric,  and  we  may  without  loss  of  general- 
ity assume  the  coiirdinate  system  in  such  a  way  that  these  cor- 
respond to  the  values  X=  dr  1.  The  sul)stitution  may  then  be 
written 


*  Consult  for  {)roof  and   historical   rc-tV'rencfs  :    Fricki'-Ivktin,    Vurleaunrjen  iiber 
die  Th'iir  ••  (b  r  (I'ltiitii'iriihi'n  Fn iirfimc  n ,   vol.    1,  jip.  -14--')',l. 


THK    BOSTON    COLLOlilirM. 


X'  -  1 

y  + 1 


X-  1 
X  +  1 


and  is  a  loxodroinic  siil)stituti()n  when  -x  =|=  0^  /3  =}=  0,  an  ellij)tic 
substitntion  when  a  =  0,  /3  4=  **,  ^uul  a  liyj)erl)()lic  ,su})stituti()n 
when  1  z^  Oj  f3  =  i).  Tlie  eorrespondini^  snbstitution  of  ,}\  is 
readily  cunipnted  t(j  he 

;»'J  =  .r^^  cosli  a  —  .i\^  sinh  0:, 

;'".,  =  .''.,  eos  /3  —  ./',,  sin  jS, 

.»■',  =  .7'.,  sin  /^  +  r*',  cos  /3, 

.'\',  ^  —  .''j  sinh  oc  -f  .r  ^  cosh  -r, 

and  the  distance  /  between  two  eorresjioiuhng  points  is  determined 
by  tlie  ('(juation 

=  cosh  0;  +  (./-|  -|-  ./'i^)  (eosh  0;  —  cos  /:?). 

J  f  7.  =  n^  eveiy  jxiint  on  the  Hne  .'■,  =  <>,  .c,,  =  0  is  fi\e(h  Hence 
an  einptie  substitution  can  not  occur  in  the  groiij)  of  the  s])ace. 
1 1"  7  =)=  0,  /  ^-  -X.  Hence  hyperboHc  or  h)XO(h'oniic  substitutions 
may  dccu!'  in  the  uroup. 

('on-idcr  next  a  j)arabobe  substitution  of  X  by  whieli  onlv  one 
vahir  of  \  is  uuallci'e<h  liv  proper  choice  ol'  the  co((rdinate  sys- 
tem tlil>  siib>titut:oii  mav  take  tlie  form 

\'  =  \  -j-  (/ 
and  tlic  coiTc-pondiuLr  sub>litutiou  of  .'•    is 


■  I 


/  ('/   —  r/  ) 


•'■-.), 


FOinrS   OF    NOX-EUC'LIDEAX    SPACE.  73 

(7  +  a  i  (d  —  a)  (nl 

(7  +  a  i  (a  —  (t)  (ta  ^ 

Tlie  tlistance  /  between  two  correspoiiding  points  is  given  Uy 
the  equation 

cosh  1=1+  7  (.'•,  -  .rj'. 


There  is  no  fixed  point  in  finite  space,  for  the  assumption  :r^  =  .r^ 
carries  with  it  the  e(piality 

^^ +  •>':=  -  1. 

We  may  however  find  corres[)onding'  points  whose  distance 
apart  is  less  than  any  assigned  quantity.  For  if  we  take  tj.  to 
represent  any  })oint,  the  cocirdinates 

represent  a  point  for  all  values  of  X  and  fi  which  satisfy  the 
relation 

X-  -j-  2x^(//,3  -  I/.;)  =  1 . 

The  displacement  /  of  the  {)oint  .r.  is  determined  by 

ad     , 
cosh  /  =.  1  -f   _^  \-(i/^  —  JJ.J-, 

and  /  can  be  made  as  small  as  we  please  by  taking  X  sufficiently 
small.  Ilence  a  parabolic  substitution  can  not  occur  in  the  group 
of  the  space. 

11 V  i/un/  Iiavc  then  c/,s'  (dloinibh'  gi-oiipx  of  a  Ch'tj'oi-d-  K/cin  .^jx/ce 
of  constant  in'f/ativc  curvature  oidy  these  which  correspond  to  groups 
of  linear  suhstitidions  of  X  irhich  are  properUj  discontinuous  u-liot 
interpreted  in.  2  and  contain  onltj  hyperholic  and  loj'odroniic  sub- 
stUutions. 

The  more  minute  discussion  of  the  Clifford-Klein  space  depends 
therefore  upon   the  knowledge  of  the  gn)Ui)s  called  by  Poincare 


74  riii-:  p,(jston  (OLLOQrir:vr. 

tlie  Kleinian  o^r()iij)s.  It  is  worth  noticing  that  wiiereas  in  the 
gronp  theory  the  greatest  attention  has  been  ])ai(l  to  Kleinian 
groups  with  elli})tic  and  paral)oli{'  substitutions,  it  is  exactly  these 
groups  which  are  of  no  interest  in  the  geometric  ])rol)leni  before 
us.  Geometry  here  waits  for  the  development  of  the  theory  of 
groups. 


SELECTED    TOPICS    IX    THE   THEORY   OF    DIVER- 
GENT SERIES  AND  OF  CONTINUED  FRACTIONS. 

By   EDWAKD   B.   VAX    VLECK. 

Part  I. 

Lectures  1-4.     Divercjent  Series. 

It  may  not  be  inappropriate  for  mo  to  preface  the  first  four 
lectures  with  a  few  words  of  a  general  character  concerning  diver- 
gent series.  These  will  serve  the  double  purpose  of  indicating 
the  nature  of  the  problems  to  be  treated  and  of  binding  together 
the  separate  lectures. 

The  problem  presented  by  any  divergent  series  is  essentially  a 
functional  one.  When  a  divergent  series  of  numbers  is  given,  its 
genesis  is  usually  to  be  found  in  some  known  or  unknown  func- 
tion. The  value  which  we  attach  to  it  is  defined  as  the  limit  of 
a  suitably  chosen  convergent  process,  and  the  elements  of  the  proc- 
ess are  the  terms  of  the  given  series  or  are  functions  having  these 
terms  for  their  individual  limits.  Most  commoidy  the  given 
numerical  series 

«„  -f  ^'i  +  ^^  +  ■  •  • 

is  connected  with  the  power  series 

(1)  a,,  +  «,.t'  +  a,.f-  +  •••, 

and  the  (piestion  thus  reduces  to  that  of  determining  under  what 
conditions  or  restrictions  a  value  may  be  assigned  to  the  latter 
series  when  x  approaches  1.  Tiie  primary  topic  therefin'o  is  the 
divergent  })o\ver  series,  and  to  this  we  shall  confine  our  attention 
exclusively. 

This  topic,  if  broadly  considered,  presents  itself  under  at  least 
fi)ur  very  different  aspects.  What  is  given  is  in  every  case  a 
power  series  with  a  radius  of  convergence  which  is  not  infinite. 
Suppose  first  that  the  radius  is  greater   than  zero  and  that   the 


7()  TirE   BOSTON  COLi:0(il'irM. 

circle  of  converijfeiice  is  not  a  natural  boundary.  Then  the  series 
detines  within  this  circle  an  analytic  function.  In  the  re<2;ion  of 
diveri^ence  without  the  circle  the  value  of  the  function  may  be 
ol)tained  l)y  the  familiar  process  of  analytic  continuation.  The- 
oretically the  determination  of  the  function  is  a  satisfactory  one, 
for  Poiiu'di't'  *  has  shown  that  the  func^tion  throughout  the  domain 
in  which  it  is  regular  can  be  obtained  by  means  of  an  enume- 
rable set  of  elements,  /*„(•''  ~  " ,)•  Pi'iK'tically,  however,  when 
IIV/o'.nY/v^.v.s'  process  is  emj)loyed  for  analytic  continuation,  the 
lai)or  is  so  excessive  as  to  render  the  ])rocess  nearly  valueless 
except  for  ])urposes  of  definition.  Hence  to-day  a  search  is  being 
made  for  a  workable  substitute.  I  may  refer  particularly  in  this 
connection  to  the  investigations  bv  Jlorcl  and  Miffdfi-Lcjjler.  As 
I  ('(tusidcr  the  work  of  the  f)rmer  to  be  both  suggestive  and 
practical,  I  have  taken  it  as  the  basis  of  my  second  lecture. 

\  second  as})ect  of  our  topic,  intimately  connected  with  the 
continuation  of  the  fimction  detined  by  (1),  is  the  determination 
of  tlic  position  and  character  of  its  singularities  in  the  region 
where  the  series  divei'ges.      This  sul)ject  is  treated  in  Lecture  '">. 

\\  Ihmi  the  circle  of  convergence  is  a  natural  boundary,  it  does 
not  apjxar  to  be  impossible,  des])ite  the  earlier  view  of  Poincare 
to  tJic  contrai'y/i'  to  discover,  at  least  in  a  certain  class  of  cases, 
an  ai)propriate,  although  a  non-aualvtic  mode  of  continuing  the 
function  aci'o>s  the  boundary  into  othei'  I'cgions  where  it  will  Ix; 
again  analvtic.  '1' he  thesis  of  llarcl \\m\  its  recent  continuation  in  the 
Ai'lii  M"//ii'iii(i/ic(i,{nvr{'\\\vv  with  some  excellent  remarks  bv  Fahri/,! 
appear  to  be  about  all  that  has  been  done  in  this  direction.  A  very 
brief'  (ii-eu~>ion  of  the  subject  will  be  given  in  the  f  »urth  lecture 
in  eoiiiKM'i  ion  with  sei'ies  of  ])olvnomials  and  of  rational  fractions. 

La-ll\-,  we  have  the  eoiniiidi'um  of  the  truly  divergent  power 
sei'ie-  —  the  serie-  which  cimsci'ges  old v  when  ./■  =  0.       It  is  upon 

•    l:.^nJ,r.,:,!i    ,1,1    Ci,,:,',,     M ,  ,1 ,' ,n,  1 1  i,; ,    ,//     I',,',,-,:,',,    V,ii.    -2     (ISSS),     p.     l-.t;,    nr    Sre 

i'.uivl  -  •/■/,/,-, ■/,  ,i,.i..n,-h,,„.,  |,.  :.:;. 

I    Tiir   I'.  iiiclii-iMiiN   iif    I'diiir.irr   ami    i'.iucl   air  imi  ariualiy  iiicnn^i^Iriil,  IjiiI  a 
new  jH.iiii  i,f  virw    i-  takrii  hv  ilic  latter. 
:  r.,.„,,,  l:. :,,/.,  v.. I.  l:^^  i  Ismm  ,  ,,,  7^. 


DIVKliCKXT  SKHIKS  AM)  CONTI  NTKD   KUACTIONS.      <  ( 

this  iiitctvstiiiu:  ))r()l)l('in  that  our  attention  will  he  especially 
focused  ill  the  first  two  lectures.  Ju  aj)j)lyin<i:  henceforth  the 
term  diverg;ent  to  ]>ower  series,  I  shall  restrict  it  to  series  having 
a  zero-radius  of  convergence. 

I  shall  offer  no  excuse  for  any  irregularity  or  incompleteness  of 
treatment.  The  admirable  treatise  })y  Borel  on  Len  Sfries  (Jltrr- 
f/enicH  (1901)  and  the  masterly  little  book  o^  IladaiiKiffl,  Jji  Sfrie 
de  Taylor  et  .son  prohiu/emcat  anah/tique  (1901),  leave  little  or  noth- 
ing to  be  desired  in  the  line  of  systematic  development,  ^^'hile  it 
is  im])ossible  not  to  repeat  much  that  is  found  in  these  books,  I 
have  also  sup])lemented  with  other  material  and  sought  to  give  as 
fresh  a  presentation  as  possible. 

Lecti'RE   1.      A.symptotic   Converr/ence. 

Few  more  notable  instances  of  the  difference  between  theoretical 
and  practical  mathematics  are  to  be  found  than  in  the  treatment 
of  divergent  series.  After  the  dawn  of  exact  mathematics  with 
Cauclij/  the  theoretical  mathematician  shrank  with  horror  from  the 
divergent  series  and  rejected  it  as  a  treacherous  and  dangerous 
tool.  The  astronomer,  on  the  other  hand,  by  the  exigencies  of  his 
science  was  forced  to  employ  it  for  the  purpose  of  com])utation. 
The  very  notion  of  convergence  is  said  hy  J^oiucare*  to  present  itself 
to  the  astronomer  and  to  the  mathematician  in  complementary  or 
even  contradictory  aspects.  The  astronomer  requires  a  series  which 
converges  raj)idly  at  the  outset.  Pie  cares  not  wliat  the  ultimate 
character  may  be,  if  only  the  first  few  terms,  twenty  for  exam])Ie, 
suffice  to  com])ute  tlie  desired  function  to  the  degree  of  accuracv 
recjuired.  ('onsequcntlv  he  judges  the  series  bv  these  terms. 
If  they  increase,  the  series  is  for  liim  non-convergent.  To  the 
mathematician  the  question  is  not  at  all  concerning  the  nature  of 
the  series  (/h  initio,  but  solely  concerning  its  ultimate  character. 

Let  me  illustrate  the  difference  bv  referrintr  to  IU'MeCf<  series 


'  "       2"/*  !  y       2(2/;  +  2)  ^  2.4(2//  +  2)  (2//  -f  -b; 

*  /yf.';  rnfth(idr.-<  nviurllis  de  In  mudiiiijiif  r('li.<tt'.  Vdl.  '2,  ]>.   1. 


78  TTIE  BOSTON  COLLOQUIUM. 

■\vliicli  is  a  solution  of  the  equation 

(2)  :r  ,  ■;  +  X  :'  +   .^-  -  n')!i  =  0. 

^     ^  (IX  (IX 

This  is  convergent  for  all  values  of  .r,  but  when  x  is  very  large 
the  series  is  worthless  for  computation  owing  to  the  rapid  and 
long'-continued  increase  of  the  terms  before  the  convergence  finally 
sets  in.  The  astronomer  and  physicist  therefore  have  been  driven 
to  use  for  large  values  of.'-  an  expansion  which  is  of  the  form  * 


i"-(^|,  +  l'  +  ^j+---) 


n  n 

+  7?.r-i  COS  x  (  />.,  +    ''+';  + 

^  X  X' 


or,  what  is  the  same  thing, 

('5 


('c"x-'^[  r;,  -f    '  +    :-f  ■ 


/>       I) 


Here  the  nudtijiliers  of  C '  and  I)  are  only  formal  solutions  of  the 
differential  ecjuation  (2).  Tn  respect  to  convergence  they  have  a 
character  exactly  opposite  to  that  of ./_,  since  for  very  large  values 
of  X  the  terms  at  first  decrease  rapidly  but  finally  an  increase 
begins.  At  this  ])()int  the  comj)uter  stops  and  obtains  a  good  ap- 
proximate value  of ,/ . 

A\'hat  is  the  significance  of  this  ?  It  is  strange  indeed  that  no 
attcmj)t  was  made  to  study  the  question  until  188G,  when  Poin- 
fdn'r  and  St 'wit j ex  \  simultaneously  took  it  u]i.  That  so  evident 
and  inii)()rtaut  a  prol)lem  should  have  ))een  so  long  ignored  by 
the  mathematician  emphasi/es  stronglv  the  need  of  closer  touch 
l)('tw«'cii  him  and  the  astronomer  and  the  physicist.  Hoth  Poincare. 
and   -SV/V///»  .N  I'egarded    the  seri(\'^  as  the  asymptotic  representation 

'  Si(>,  for  ('x:irn)ilc,  (iray  and  Matlifw's  '1  rratlfr  an  Besf,il  ]''iiii(iii)n.t,  chap,  4. 
i  Ann   Mall,.,  V(.l.    S,  ,,.  '^',i:,  IT. 

:  77,-.;.,  An,,.  ,1.  r  Kr.  .\„,:,  stT.  r.,  v..i.  :\,  \k  201. 


DINEIJGENT  SEllIP^S  AXI)  (ONTIXrKD  FRACTIONS.       7it 

of  one  or  more  functions,  ^\^llil('  tlu'  latter  writer  studied  care- 
fully certain  divcru;ent  series  of  special  importance  with  the  object 
of  obtaining-  from  the  series  a  yet  closer  appro.ximation  to  the 
function  l)y  a  species  of  iuterpolation,  J^oinccrf  developed  the 
idea  of  asymptotic  rej)resentation  into  a  general  theory. 

To  e.\i)lain  this  theory  *  and  at  the  same  time  to  develop  certain 
aspects  scarcely  considered  by  Poinrdre,  I  shall  start  with  the 
genesis  of  a  Taylor's  series.  Take  an  interval  (0,  a)  of  the  posi- 
tive real  axi?*,  and  denote  by  _/(.'■)  any  real  function  which  is  con- 
tinuous and  has  n  +  I  successive  derivatives  at  every  point  within 
the  interval.  Xo  hypothesis  need  be  made  concerning  the  char- 
acter of  the  function  at  the  extremities  of  the  interval  except  to 
sujipose  that  _/"(■'-'),/'(:'.'),  •  •  •,  /'"  {j')/'ii  I  have  limiting  values  a^^,  a^, 
■  ■  ■,  (/^  when  .>'  approaches  the  origin.  Thus  the  function  at  any 
point  within  the  interval  will  be  represented  by  Taylor's  formula  : 

/(.,.)  =  ..,,  -f  r/,r  -f  a^^  +  •  •  •  +  ax^'  +  ^,,  ^  yy^  /-"^ ' '  (^-'O 

(0<^<1). 

If  the  function  is  unlimitediv  ditiferentiable  and  limiting^  values 
of _/'  "  (■'■)//'  I  exist  tor  all  values  of  n  when  .r  aj)proaches  0,  the 
number  of  terms  in  the  formula  can  be  increased  to  anv  assigned 

*  o 

value.      Thus  the  function  gives  rise  formally  to  a  series 

tuii(piely  determined  by  the  limiting  values  of  the  function  and  its 
derivatives. 

The  converse  conclusion,  that  the  series  determines  iniiquely  a 
function  fultilling  the  conditions  above  imposed  in  some  small  in- 
terval ending  in  the  origin,  can  not,  however,  be  drawn.  This  is 
not  even  tiie  case  when  the  series  is  convergent.  Suppose,  for 
exam])le,  that  c    =  0  for  all   values  of  /(.     Then    in    addition    to 

■■  Cf.  reaiio,  Atti  ileWi  R.  Accail.  (IpIUi  Sci.nze  di  Torinn,  vol.  27  (1891  ),  p.  40  ; 
reproduced  a<  Aiili<iiif/  III  (  "  t'eljer  die  Taylor" sclie  Foi-inel"'  )  in  (ieiiucclii- 
Peano's  DiijT'  ntinl-  nail  fnfeffrdl-Iiechiiiiiit/.  p.  ,3")'.t. 


so  TiiK  EosTox  coLL()Qrir:\r. 

f(.r)  =  0  we  have  tlie  finietions  (~^  '',  r~'  ""',  ■  ■  ■  ,  which  fulfill  the 
assij^ned  conditions.  They  are,  namely,  nnlimitedly  diiferentiable 
within  a  positive  interval  terminating  in  the  origin,  and  when  x 
approaches  the  origin  from  within  this  interval,  the  functions  and 
their  derivatives  have  the  limit  0.  ]"^rom  this  it  follows  imme- 
diately that  if  values  other  than  zero  he  prescribed  for  the  a  ,  the 
function  will  not  be  imiquely  determined,  since  to  any  one  deter- 
mination we  may  add  constant  multiples  of  r"''-',  c~'  -^"j  ■  •  •  . 

Inasmuch  as  the  corres])ondencc  between  the  function  and  tlie 
series  is  not  reversibly  unique,  the  series  can  not  be  used,  in 
general,  for  the  computation  of  the  value  of  the  generating  func- 
tion. Xevertholess,  although  this  is  the  case,  the  series  is  not 
without  its  value.  For  consider  the  first  //(  terms,  //(  ])eing  a 
fixed  integer.  If  x  is  sufficiently  diminished  in  value,  each  of 
these  terms  can  be  made  as  small  as  we  choose  in  comparison  with 
the  one  which  })recedes  it,  and  the  series  therefore  at  the  begin- 
ning has  the  aj)pearance  of  being  ra])idly  convergent,  even  though 
it  be  really  divergent,  l^vidently  also  as  x  is  decreased,  it  has 
this  a])pearance  for  a  greater  and  greater  number  of  terms,  if  not 
throughout  its  entire  extent.  Xow  l)y  hy]H)thesis  the  generating 
function  was  unlimitedly  differentiable  within  the  interval,  and 
the  successive  derivatives  are  consequently  continuous  within  (0,  (t). 
Hence  if  the  interval  is  sufliciently  contracted,  _/'"''  (.'■)/(//)  +  1)1 
can  be  niadc^  as  nearlv  (Mpial  to  '/^  .,  throughout  the  interval  as  is 
desir<'(l.      We  have  then  for  the  I'emainder  in  Tavlor's  formula  : 

/■ '(Ox) 

in  wliich  e  is  an  arbitrarilv  small  positive  (|ua!ititv.  ( 'ousccpu'iitly 
if  the  lii'-t  ///  -|  I  terms  of  the  scries  should  be  used  to  comjiute 
the  vahie  of  the  generating  function,  the  ei'ror  committed  would 
be  ajjprnxiinately  e(|Mal  to  the  next  term,  jirovided  .'•  be  tai'Cen  suf- 

tieielltlv    ~lliall. 

in  the-e  eoii-iderat  ion-  there  is,  of  course,  nothing  to  indicate 
when  ,'•  i-  sunieiently  small  for  the  piirj)ose.       If  the   result    holds 


DIVKKGFAT  SKRIHS  AXI)  CON'TI  XUKI)  FRACTIONS.       <S  1 

siiniiltaneously  for  a  lari^e  number  of  consecutive  values  oi'  id,  the 
l)est  possil)le  value  f  )r  the  function  consistent  with  our  informa- 
tion would  evidently  i)e  obtained  by  carrying  the  computation 
until  the  term  of  least  absolute  value  is  reached  and  then  stoi)ping. 
Herein  is  j)r()bably  the  justification  f  >r  the  practice  of  the  com- 
j)uter  in  so  doing. 

Equation  (4)  which  gave  a  (iuiit  to  tJic  error  in  sfoj^pliif/  iritli 
fhr  (lit  -\-  1)///  fcriii  shows  also  that  t/ils  /imif  (/roirn  .^uudler  c.s  x 
'tiniinis/ics.  Since,  furthermore,  by  increasing  in  sufficiently  the 
(//(  -f  2)th  term  of  (1)  may  be  made  small  in  comparison  with  the 
(/;/  -f  l)th  term,  it  is  clear  that  on  the  whole,  as  .r  (Ilminishes,  we 
inusf  tdhf  a  (jrcdfcr  (ind  r/reater  nuiaber  of  terni.s  to  secure  the  best 
(ipproxiinotioii  to  the  fu iictioii.  These  two  facts  may  be  comprised 
into  a  single  statement  by  saying  that  the  aj)j)roximation  given 
by  the  series  is  of  an  asymptotic  charac^ter.  This  will  hold 
whether  the  series  is  convergent  or  (b'vergent. 

This  notion  can  beat  once  embodied  in  an  e(jUation.  l-'rom  (4) 
we  have 

,,,,,/(..•) -"„-",»^ „/:: 

=  lim  ^'";;,:-  =0  (.;=!,  2,...). 

4'his  (Mjuation  is  an  exact  e(juivalent  of  the  two  [)roj)erties  just 
mentioned  and  is  adc)j)ted  by  Po'incure  *  as  the  definition  of  asynij)- 
totic  convergence.  ^lore  explicitly  stated,  the  series  (1)  is  said 
I)y  liim  to  r())resent  a  function  /(r)  asymptotically  when  equation 
(o)  liojds  for  all  values  of  in. 

It  will  be  noticed  that  this  definition  omits  altogether  the 
assum])tioiis  concerning  the  nature  of  the  function  with  which  we 
started  in  deriving  the  series.  Xot  onlv  has  the  reijuirement  of 
unlimited  ditferentiabilit v  within  an  intei'S'al  been  omitte(l  but  the 
existence  of  right-hand  limits  for  the  derivatives  as  ,<■  approaches 
the  origin  is  not  even  postulate*!.      If  the  value  o ^^  be  assigned   to 

■•Foe.  cit. 


82  THE  BOSTOX  COLLOQUIUM. 

the  function  at  the  origin,  it  will  have  a  first  derivative,  a^,  at  this 
point  bnt  it  need  not  have  derivatives  of  higher  order.* 

The  exclusion  of  the  re(juirement  of  differentiability  has  un- 
doubtedly its  advantages.  It  enlarges  the  class  of  functions  which 
can  be  represented  asynn)totically  by  the  same  series.  It  also 
simplifies  tlie  a})pllcation  of  the  theory  of  asymj)totic  representa- 
tion, and  this  is  perhaps  the  chief  gain.  The  results  of  Voiacare^s 
theory  can  readily  be  surmised.  The  sum  and  product  of  two 
functions  re])resented  asymptotically  by  two  given  series  are 
represented  asymptotically  by  the  sum-  and  prochict-series  respec- 
tively, and  the  (juotient  of  the  two  functions  will  be  represented 
corres})ondingly,  provided  the  Constant  term  of  the  divisor  is  not  0. 
Also  if  _/"(.'•)  is  any  function  represented  by  the  series  (1),  whether 
convergent  or  divergent,  and 

(/)(.,.)  =  h^^  +  b^,v  +  />,r-  +   .  .  . 

is  a  second  series  having  a  radius  of  convergence  greater  than  \c(^\, 
the  asymptotic  representation  of  (/)[/'(.'■)]  will  be  the  series  which 
is  obtained  from 

^,  +  '\{".  +  "v''  +  •■•)  +  ^^.("o  +  <'i'''  +•••)'+••■ 

by  rearranging  the  terms  in  ascending  ])owers  of  :'■.  J^astly,  the 
integral  of_/'(.'')  will  have  for  its  asymjitotic  rei)resentation  the 
term  l)y  term  integral  of  (1  ).  But  the  correspondence  of  the  func- 
tion and  series  may  b(!  lost  in  dltferentiation,  for  even  if  the 
function  |)ermits  of  differentiation,  its  derivative  will  not  neces- 
sarilv  be  a  function  having  an  asymptotic  pow(>r  sei'ics.  lOxamples 
of  this  kind  can  be  i-eadiiy  gi\-en.t 

* 'I'lic  nrdinai'v  ilflliiitiDii  nf;ni  ;itli  licrivativc  is  luTi' assumed.  If,  however, 
\vr  dfliue  till.'  sct'Diiii  ilt-rivativc  liv  the  i-\]>i-fssi(iii 

J-     0  '" 

and  till-  liiijlicr  ilt-ri  val  i  \c>  in  similar  t'a>liinn,  the  fiiiict  inn  mn-t  liavi-  (Ut!  vat  i  vi'S 
nf  all  onl.T^. 


DIVEIKJKXT  SEKIES  AND  CONTI Xl'KD   FKACTIOXS.       83 

This  failure  is  on  many  accounts  an  unfortunate  one.  If  a 
furtlier  development  of  PoiiK'(ii-e\s  theory  is  to  be  made  —  and  this 
seems  to  me  both  a  possibility  and  a  desirability  —  his  definition 
})robably  should  be  restricted  by  requirint^  {</)  that  the  function 
corresponding  to  the  series  shall  be  nnlimitedly  dilferentiable  in 
some  interval  terminating  in  the  origin,  and  (6)  that  the  deriva- 
tives of  the  function  should  correspond  asymptotically  to  the 
derivatives  of  the  power  series.  These  demands  are  satisfied  in 
the  case  of  an  analytic  function  defined  by  a  convergent  series  and 
seem  to  l)e  indispensable  for  an  ade(|uate  theory  of  divergent 
series.""' 

Thus  far  we  have  considered  asymptotic  representation  only  for 
a  single  mode  of  approach  to  the  origin.  Suppose  now  that  an 
analytic  function  of  a  complex  variable  x  is  represented  by  (1)  for 
all  modes  of  approach  to  the  origin,  and  let  a^  be  the  value  assigned 
to  the  function  at  this  })oint.  Then  if  the  function  is  one- valued 
and  analytic  about  the  origin,  it  must  also  be  analytic  at  this  point 
since  it  remains  finite.      Hence  the  series  must  be  convergent. 

The  case  which  has  an  interest  therefore  is  that  in  which  the 
asymptotic  representation  is  limited  to  a  sector  terminating  in  the 
origin.  Suppose  then  that  (1)  is  a  given  divergent  series,  and  let 
a  function  be  sought  which  fulfills  the  following  conditions  :  {(() 
the  function  shall  be  analytic  within  the  given  sector  for  values  of 

*  These  requirements  are  fortnulated  from  a  mathematical  standpoint  with  a 
view  to  extendinji;  the  tlieory  of  analytic  functions,  and  doul)tless  will  be  too 
stringent  for  various  astronomical  investigations.  Prof.  E.  W.  Brown  suggests 
that  for  such  investigations  the  conditions  might  perhaps  be  advantageously 
modilied  by  making  the  requirements  for  only  m  derivatives,  m  being  a  number 
whicli  varies  with  x  and  increases  indefinitely  upon  approach  to  the  critical  point. 
He  also  points  out  the  difficulties  of  an  extension  in  the  case  of  numerous  astro- 
nornic:\I  series  wlilcli  have  tiic  form/(/,  t)  -=  a^^  ~  tt^r  -f-  'i.,.>''^  "h  '  '  'j  where  ((^  is  a 
function  of  .'■  and  t,  cf/ct  lieinga  convergent  series.  Foincar(5's  definition  is  how- 
ever still  applicable. 

Oftentimes  in  celestial  mechanics  the  only  information  concerning  the  func- 
tion sought  is  afl'orded  in  tiie  ap])roximation  given  by  the  asymptotic  series.  An 
objection  to  Poincare's  definition  is  tliat  it  presupjioses  a  knowledge  of  the  func- 
tion souglit,  for  example,  that  lim  /(/)  c,,,  when  /  0.  As  a  matter  of  fact 
the  properties  are  often  unknown.     See  in  this  connection  p.  SO  of  these  lectures. 


.S4  THE  BOSTON  COLLOQX^IUM. 

X  which  are  suflieiently  near  to  the  origin;  (/>)  it  shall  be  re})re- 
sented  asymptotically  by  the  given  series  within  the  sector,  whether 
inclnsive  or  exclusive  of  the  boundary  will  remain  to  be  deter- 
mined ;  ((')  the  asymj)totic  representation  shall  not  be  valid  if  the 
angle  of  the  sector  is  enlarged,  80  far  as  I  am  aware,  the  exist- 
ence of  a  function  or  of  functions  which  meet  these  requirements 
has  never  been  demonstrated,  though  it  seems  likely  that  they  in 
general  exist.  It  is,  however,  very  possible  that  the  sector  nuist 
be  restricted  in  ])osition  as  well  as  in  magnitude.  It  may  be 
found  necessary  to  require  that  the  interior  of  the  sector  shall 
not  include  certain  arguments  of  .r ;  for  example,  in  the  case 
of  the  series  Im  I.i'"'  *  the  argument  0,  for  which  the  terms 
have  all  the  same  sign,  j  If  this  be  true,  the  sector  will 
very  j>rol)ably  liave  two  such  arguments  for  its  l)()undaries. 
When  there  is  a  function  which  satisfies  the  conditions  im- 
posed, it  can  not  be  uni(|ue.  For  clearly  e~'  '',  r'~'  ■^',  e^'  ^^,  ■  ■  •, 
within  certain  sectors  of  angle  tt,  'Itt,  ott,  •  •  •,  have  an  asymptotic 
series  in  which  each  coefficient  is  0.  If,  then,  any  function  has 
been  obtained  satisfying  the  conditions  stated,  one  or  more  of  these 
exponentials,  after  multij)lication  by  suitable  constants,  may  be 
addeil  to  the  function  without  destroving  its  j)roperties.  Hence 
if  a  <livergent  sei'ies  is  to  I'epresent  a  function  unifpiely,  su})))le- 
mentary  (•()nditions  iiuist  be  imjjosed.  The  nature  of  these  condi- 
tions has  not  yet  been  ascertained.  ;}; 

Fu  cln-iiig  the  general  discussion  a  simple  extension  of  the 
notidii  i>f  asymj)ti>tic  ('(invergcnce  >liouId  be  mentioned  wiiich  is 
iieres-arv  for  the  applications  to  follow.  /■'(.'•)  is  said  to  be  repre- 
sented asvinptoticall V  bv 

'■'(•'■):"„  -f  '',•'•  +  ",•'•'  +  •  •  • ; 

"    1  lii^  ~riii'>  i>  (i  i-ciis^i'il  ill  till'  next  Icrtiirf. 

'  llnrd  Inr,  ,■//.,  |i.  ."I'll  ill  h  i>  (.'x  |M  i-ii  ii  111  iif  roiiicufc''^  I  lici  >rv  sc'fiiis  to  make 
t  lie  ili-liii  lie  -taiciiHiii  I  liiit  1  lure  ii  re  ;i  I'LTM  iiiriit~  for  which  ii<i  con'opuniiinL;  fuiK'- 
1  i'.ii  IX  i^t'-.   I  lilt    I   ,1111  1 111:1  hh-  In  111  1(1  .■my  I  III  ml"  of  t  he  slalciiiciit. 

:  In  ihi-  i-iiiiiirciidii  MT  pp.  S'.i-'.fJ  111'  Iliircl',-  arlicic,   .!((;).  </«•/'/•>.  .\'>r.,  .ser.  'A, 

V..!.  ii;    i-'.i'.i  i. 


])1VE1{(;kxt  series  and  contixi'ed  fk actions,     s-j 

when    tlie   series    in    ]);irentliesis   gives    such   a    representation    of 

The  applications  of  J\>iiic<ires  tlieory  liave  been  made  cliieHy 
in  the  province  of  ditferential  etjnations*  where  divergent  series 
are  of  very  common  occurrence.  We  will  take  for  examination 
the  class  of  ecjuations,  of  which  tlie  theory  is  })erha])s  the  most 
widely  known,  the  homogeneous  linear  dilferential  equation  with 
polynomial  coefficients  : 

(«)     ^„(^')  2' + ^''•-•('^)  t-^ + •  •  • + ^'^'^^' = *'• 

This  is,  in  fact,  the  class  of  e(}uations  to  which  Poiitcare  first 
applied  his  theory, t  but  his  discussion  of  the  asymptotic  repre- 
sentation of  the  integrals  was  limited  to  a  single  rectilinear  mode 
of  approach  to  the  singular  point  under  consideration.  The  de- 
termination of  the  sectors  of  validity  for  the  asymptotic  series 
lias  been  made  by  Jlorn^X  ^^'l^^*  i'^  •'  number  of  memoirs  has  care- 
fully studied  the  application  of  the  theory  to  ordinary  differential 
equations.  § 

As  is  well  known,  the  only  singular  points  of  (G)  are  the  roots 
of  P_l'i')  and  the  })oint  .r  =  oc.  For  a  regular  singular  point  |j  we 
have  the  familiar  convergent  expressions  for  the  integrals  given 
by  Fuchs.  Consider  now  an  irregular  singular  point.  By  a  linear 
transformation  this  point  maybe  thrown  to  oc,  the  equation  being 
still  kept  in  the  form  ((>).  Suppose  then  that  this  has  been  done. 
If  P.^  is  of  the  />th  degree,  the  condition  that  .r=  oc  shall  be  a 
reticular  singular  p(Mnt  is  that  the  degrees  of  J*  ,,  I^  .,,••■,  P„ 
shall  be  at  most  e(|ual  to  y>  —  \,  p  —  '2,  ■■■,[>  —  /*,  res])ectively. 

For  an  irregular  singular  point  some  one  or  more  of  the 
degrees  must  be  greater.  Let  //  be  the  smallest  positive  integer 
for  which  the  degrees  will  not  exceed  successively 

*  Jn  iiddition  to  the  memoirs  cited  below  I'oincai-^'s  L-'s  metluxlfs  iioiirc/lr.-^  d-  In 
riv'ctinique  celeMe  and  various  memoirs  by  Kneser  may  l)e  coiisnlte<l. 

\Acta  Mdth.,  vol.  8  (18.S6),  p.  803.     See  also  Amer.  Jour.,  vol.  7  ( ISS-i),  p.  -JO;}. 

iM'ith.  Ann.,  vol.  oO  (1898),  p.  525. 

?  See  various  articles  in  CVc/Ze'^'  ,/';(()•;("/ and  the  ^[(^(}i<'ln(it^.■iche  Annalen. 

!!  Stelle  der  Px'Mnnndheit. 


86  THE  BOSTON  COLLOQUIUM, 

y.  +(A-1),     ^.  +  2  (//-I),     y.  +  3(A-l),  .... 

The  number  h  i^s  called  the  rank  of  the  siuirular  point  oc,  and  the 
ditl'erential  e(juation  can  be  satisfied  formally  i^y  the  series  of 
T/ioiii(ie  or  the  so-called  normal  series  : 

(7)  ,S,  =  /""^"-'"^-"— 'V(6^+  *^'  +  %  +  •■■) 

('■=1,2,  ...,,,)■ 
Unless  certain  excejitional  conditions  are  fulfilled,  there  are  n  of 
these  expansions,  and  in  general  they  are  divergent.  To  simplify 
the  presentation  let  us  confine  ourselves  to  the  case  for  which 
//  =  1.  Then  at  least  one  of  the  polynomials  succeeding  P^  will 
be  of  the  /^th  degree,  and  none  of  higher  degree.      Place 

r^=A^j^-\- 11,'"-'  +  ■■■, 

and  construct  the  equation 

(8)  Aa"  +  A,^^^o.'-'  +  ...  +  A^=i\ 

The  n  roots  of  this  equation  are  the  //  quantities  a.  which  aj>])ear 
in  the  exjionential  comj)onents  of  the  S.. 

As  a  })articular  illustration  of  the  class  of  e<jUations  under  con- 
sideration, 7>r.v.sv'/'.s  equation  (  ]^q.  (2)  )  may  be  cited,  llei'e  the 
])oint  QC  is  of  rank  1,  the  characteristic  ('(juation  is 

yl,a-  +  A  ,0-.-  -f  A,  =  a-  -^  \  =(\ 
with  the  roots 

^1  =  -  ',      ^,  =  +  ', 

and  the  two   T/kjukk an  integrals  are 

;!,  =  '''■ ■'(':.+  '"'  ■+  ■ 


I)IVI-:RGEXT  series  and  COXTIXI'ED  FRACTIOXS.       ST 


ill  which  /),,  p.,  are  yet  to  bo  ascertained.  After  this  has  been 
<l()iie,  the  coethcients  of  (0)  can  be  (letermined  by  direct  substitu- 
tion in  (2). 

To  avoid  comi)lications  we  will  assume  that  the  n  roots  of  the 
characteristic  e(|uation  (S)  are  all  distinct,  also  that  the  real  })arts  of 
no  two  roots  are  equal.  ]\[ark  now  in  the  complex  ])lane  the  points 
aj,  a„  ■  ■  •,  a  ,,  and  draw  from  them  to  intinity  a  series  of  ])arallel 
rays  having  .such  a  direction  that  no  one  of  the  rays  with  its  pro- 
loniration  in  the  opposite  direction  shall  contain  two  or  more  of 
these  points.      Finally  surround  the   })oints  a.  with   small  circles, 


so  that   we  shall  have  the  familiar   loop  circuits   for  the  ])atlis  of 
intef2:ration  of  the  integrals  which  we  now  proceed   to  form.      Put 


(10) 


1,=  fr-r^dz 


{1=  I,  ■■;  n), 


in  which  v.(z)  is  a  function  to  be  subsequently  fixed.  In  order 
that  the  integral  may  have  a  sense,  x  will  be  so  restricted  that  the 
real  part  of  z.v  shall  be  negative  for  the  rectilinear  parts  of  the 
loop  circuits.  We  can  then  so  determine  r.{z)  that  r].  shall  be  a 
solution  of  (G). 

For  this  purpose  substitute  7;.  for  //  in  (G).      A  reduction,  based 
on  the  integration  of  (10)  by  parts,*  gives  for  v.(^z)  the  equation 

(11)     {A/^  +  .1„_,."-  +  ■  • .  +  A^)  2:  +...+  ().  =  0, 

This  is  known  as  Jjdphtcex  transformed  equation.  While  the 
original  equation  was  of  the  »th  order  with  coefficients  of  the  y>tli 

*  (,'f.  Piciird's   Truitv  d' Anali/tie,  vol.  3,  p.  38o  fT.,   or   Poiiican',   A>nrr.   Jour., 
vol.  7  (ISS-',),  p.  217  fV. 


88  THE  BOSTON  COLLOQUIUM. 

clcu'i't'c,  tlic  transform  is  of  the  y>tli  order  with  cocfficioiits  of  the  nth 
clei>ree.  Its  sinoiilar  points  in  the  finite  phxne  are  the  roots  of  the 
first  eoeflieient  of  (11),  whieli  is  identical  with  the  left  hand  nieni- 
hei'  of  (S).  I'^irtherniore,  an  inspection  of  ( 1  1 )  shows  immedi- 
ately that  each  of  these  sinti;nlar  j)oints  a.  is  regnlar,  and  the 
e.\])onents  whieli  heloni;-  to  it  are 

in  which  p.  is  the  exj)onent  of  x,  hitherto  inidetermined  in  (7). 
Hence  if  l3.  is  not  an  integer,  there  is  an  integral  oi"  (11)  having 
the  form 

(,_. j3,(/., +  /.(,_  a,) -f  /,(.:- a,)^ +...), 

which,  when  continued  analytically,  can  he  taken  as  the  fnnction 
v..      Thus  foi"  the  solution  of  (())  we  obtain 

7/.  =  j^ .-(.  -  a/.(/-,  -f  /•,(-  -  0=,)  +  ■  •  -y-:. 

]f,  finally,  n/.^  -\-  i/J.r  is  suhstituted  for  z  tiie  integral  becomes 

where  the  transformed  ])ath  of  integi'ation  is  a  looj)  circuit  which 
encloses  the  origin  of  the  //-plane,  the  rectilinear  poi'tion  of  the 
])ath  King  in  the  half  plane  for  which  the  I'cal  part  of//  is  negative. 
We  have  thu-  reached  a  solution  of  the  differential  e(|Uation 
under  the  foi'm  of  an  impro])er  integral  of  a  convei'gent  series, 
"^riie  integi'ation  of  (12)  teiMu  by  term,  which  is  a  j)urelv  formal 
process,  gives  at  once  the  normal  integral  N  of  (7j,  in  which 


r,   „  =  /;,    j  r"f'-^-,lii 


'\\\f  asymptotic   character   of  S.  can    be  quicklv   demonsti'ated.' 
I'^or  let  n"/'  ill)  denote  the  remainder  aff<'i"  /;  terms  of  the  series 

/•„  +  /■,"  +  /'V^'+  ■■•■ 

Then 

*  llniii,  Inr   r//,,  ,,r  An,,  Mnffi.,  vol.  ■2\  (I'.HIlj,  pp.  -IW  IT. 


i)i\"i:K(;i:x'r  si:i;iks  and  cox'riNrKi)  fkactioxs.     Si) 


n    [  ■  \<h 


4/-,. 


Since  tlio  integ-ral  in  the  n'u'lit  hand  niemher,  taken  ah)nu'  the  loop 
circuit,  can  be  shown  to  remain  finite  when  .'■  =  zc ,  we  have 


But  this  is  the  statement  of  P(nnc<ire!<  definition  of  asymptotic 
convergence  for  .'•  =  ac. 

1  liave  sketched  this  lengthy  j)r()cess  in  some  detail  because  it 
is  a  thoroughly  ty})ieal  one  and  indicates  the  j)resent  status  of  the 
theory  of  asymptotic  series.  It  will  be  observed  that  the  follow 
ing  course  is  })ursued  : 

1.  First,  it  is  discovered  that  the  diti'erential  equation  permits 
of  formal  solution  by  a  certain  divergent  series. 

2.  By  some  independent  ])rocess  the  existence  of  an  actual  solu- 
tion is  ascertained  which  ])ermits  formally  of  exj)ansion  into  the 
series.  T.suallv  the  solution  is  found  under  the  form  of  an  inte- 
gral, and  ILirn  has  applied  the  theory  chiefly  in  cases  in  which 
solutions  of  this  form  were  known.  (Lately,  however,  he  has 
used  solutions  obtained  from  the  diflPerential  er|uation  by  the 
process  of  successive  ap})roximation.*) 

'■>.  The  asym])totic  character  of  the  series  is  then  argued  and, 
finally,  the  sector  within  which  this  representation  is  valid  is 
determined. 

The  status  of  the  theory  thus  exhibited  seems  to  me  an  unsat- 
isfactory and  transitional  one.  It  is  to  be  hojxxl  that  ultimatelv 
the  theory  will  be  so  developed  that  the  mere  existence  of  a  diver- 
gent power  series  as  a  foruuiJ  solution  of  the  differential  equation 
will  l)e  suillcient  for  the  immediate  affirmation  of  the  existence  of 
one  or  more  solutions  which  are  analytic  functions  with  certain 
sj)ecificd  proj)erties. 

''■  ^f.ltll.  Ann.,  vol.  .')!  (lS98),p.  :U(!.  In  CrI/r  .■<  JonyiKfl,  vq].  118(1807), 
still  another  mctluxl  is  used  for  obtainin";  the  solutions. 


90  THE  BOSTOX  COLLOQUIUM. 

It  remuiiis  yet  to  fix  tlie  sectors  within  wliicli  the  solutions  rj. 
can  be  re})resented  asymptotically  by  the  normal  integrals.  These 
sectors  have  been  specified  by  Iloni  *  in  the  following  manner. 
Let  straight  lines  be  drawn  from  each  singular  point  a.  to  every 
other  point  and  produce  each  joining  line  to  infinity  in  both  direc- 
tions. A  set  of  lines  will  be  thus  fixed,  radiating  from  the  point  oo. 
Let  their  arguments,  taken  in  the  order  of  decreasing  magnitude, 
be  denoted  by 

&)|,    ft).,,    •  •  •  ,   ft)  ,   ft)       I   =  "^1  —  "^j    •  ■  ■  ,   <^..,.  =   (^,.  —  TT. 

Suppose  now  that  the  argument  of  the  rectilinear  part  of  the 
path  of  integration  for  ?;.  in  the  plane  of  z  lies  between  &>^_j  and 
&)p.  Then  ?;.  is  represented  asymj)totically  by  .^'.  for  values  of  the 
argument  of  a;  between  7r/2  —  &)       and  ttJ'I  —  «„+,.•! 

To  the  general  solution  of  (6),  c^rj^  -f  r,,?;.,  -f  .  . .  -f  e  ?;  ,  there 
corresponds  the  divergent  expansion 


<'^^,  +  •■■  +  r.'^.  =  '•/'"".'■''> K',  +  - '" '  +    '•..'-'  +  •  •  •  + 


1 


3'  X 

(18) 

■  ■  ■  -\-e  e°"^fp"  I  ('  +     "' '  +     ";-  4-  ■  •  ■  )• 
\    "  X  x-  J 

Here  the  real  ])arts  of  two  ex])onents,  a.x  and  ax,  are  equal  only 
when  ar(j{a.  —  a.)x  is  an  odd  multijile  of  7r/2 ;  that  is,  when  (irgx 
is  eijual  to  7r/2  —  to.  (/  =  1,  •  •  •  ,  2/').      Su]>pose  then  that  for 

we  so  assign  subscrij)ts  to  the  a    that 

Then  all  the  integrals  f )r  which  (\  =j=  0  have  in  common  the 
asvnij)totic  scries  ('|>V  ,  while   those   for  which  c^  =  c.,  =  •  ■  ■  :=  r.^, 

*  Horn,  Mnth.  Ami.,  vol.  .",()  (IS'JS),  J,.  r,;il. 

t  In  ciTtriin  (■;i~('-  the  ;isyiiii)totic  rcprcsciitatinii  iii;iy  he  valiil  fur  a  f^reatcr 
raii^i'  iif  values  dt'  tlic  ai'fxiiiiifiit  (if  .1,  as  in  the  case  of  Hosscl's  iM|uatinn  discusst'd 
below. 


DIVEIUIEXT  SEKTKS  AND  CONTTNUKI)  FRACTIONS.      !>1 

c.  =1=  0,  are  rcjiresciitt'd  by  r.S^.  Tlius  it  apjicars  tliat  between  tlie 
arguments  considered  -V^  is  the  only  one  of  the  /;  asymptotic  series 
S.  which  defines  a  sohition  of  the  differential  equation  (6)  uniquely. 
Chanues  in  the  asym])totic  series  re])resenting  a  solution  may 
occur  from  two  causes,  either  because  .r  passes  through  one  of  the 
critical  values  above  mentioned  for  which  there  is  a  change  in  the 
dominant  exj)onential  in  (13),  or  because  of  a  sudden  alteration  in 
the  values  of  the  constants  c.  for  certain  values  of  the  arirument. 
This  can  be  made  clear,  in  conclusion,  by  illustrating  with  />f'.s\sr/',9 
equation.*      For  this  equation,  as  we  saw, 

aj  =  —  /,      a.,=  +  i, 
and  hence 

Stt  tt 

Also  since  Laplace's  transform  for  the  i)articular  case  before  us  is  f 

72  7 

(,^+l)'^^^+3.J^^+(  )v  =  0, 

the  exponent  p^  for  either  of  the  two  singular  points  z  :=  ±  i  has 
the  value  —  h  Accordingly  the  series  (13)  for  c^rj^  -j-  c.,ri.,  may  be 
written 

cv"-^-(c^  + §  +  §+•■ 

+  /),-■'«;-■.(/)„  +  :^J-  +  ^=  +  . , .  ^  =  Cr-(,r)  +  i)T-(,r), 

as  previously  given  in  (3).  If  the  imaginary  part  of  x  is  nega- 
tive, CC(:r)  is  the  dominant  term  in  (3)  and  gives  the  asymptotic 
representation  of  the  general  solution,  c^r/^  -f  c.,t].,.  On  the  other 
hand,  if  the  imaginary  part   is    positive,  the  dominant   term    is 


*  A  brief  but  very  interesting  discussion  is  given  in  a  letter  of  Stokes  in  tiie 
Acta  Math.,  vol.  26  (1902),  pp.  393-397.     Compare  also  ?3  of  Horn's  article, 

Math.  Ann.,  vol.  50  (1898),  p.  525. 

■f  M"ih.  Ayvn.,  vol.  50,  p.  539,  Eq.  B'. 


02  THK   r.OSTOX   COLLOQUir^r. 

]>  I '(.'•).  The  ('li;uio:es  in  the  vahics  of  C  and  I)  take  phiec  only 
when  arg- ,'•  ])asses  throuo'h  the  vahies  {'In  -\-  \)tt j'l.  Tlien  tlie 
enefficieut  of  the  dominant  tei'ni  remains  unaltered,  while  the  coeffic- 
ient of  the  interior  term  is  altered  by  an  amount  ])ro[)ortional  to  the 
coetlicient  of  the  dominant  term.t  A\'e  conclude,  therefore,  that 
in  o-eneral  the  asymptotic  series  f  tr  any  solution  of  Bcs^eV-^  equa- 
tion changes  al)ruj)tly  for  values  of  the  aro-ument  con<>;ruent  ■with 
()  (mod  it).  Furthermore,  the  series  can  not  be  valid  for  a 
greater  range  of  values  of  the  argument  unless  ^\•hen  arg  x  =  0, 
either  J)  =  0  or  ('  =  0.  In  the  ibrmer  case  we  liave  a  particular 
solution   C'?;,  which  is  re})resented  by  the  scries  Cl\,r)  for 

—  TT  •<  arg  .(■  <;  27r, 

and  in  the  latter  case  a  solution    J>7].,  re])resented   by  J)^\,r)  for 

—  '27r  -^  arg  ,<■  -^  tt. 

Thus  from  the  iuHnitely  many  solutions  of  7>V'.s'.sr/'N  e(]Uation  having 
the  common  asymptotic  rej)resentation  ^7 '(:'•)  and  /''I'(.^')  respec- 
tively, these  two  solutions  can  be  singled  out  by  the  requirement 
tliat  the  asym])totic  rej^resentation  shall  have  the  maximum  sector 
of"  validity. 

I.i;('rri;i',  '2.      The  Aj/j/licafion  of  Iiifrf/ri//s  to  Ilircrrjott  Scricx. 

In  the  tii'st  lecture  a  divergent  s(>ries  was  contiected  with  a  grouj) 
ol' t'unctions,  ibi-  which  it  afforded  a  connnon  asymptotic  rej)rcscn- 
tation.  In  the  j)i"esent  lecture  I  shall  treat  of  methods  wliicli 
have  bfcii  u<cd  to  derive  a  function  uni(iuely  from  the  series. 
To  c-tabli^h,  whenever  jtossible,  such  a  uni(jU<'  connection,  to 
dcvolo])  the  properties  of  the  function,  and  to  determine  the  laws 
and  conditinu^  under  wliich  the  series  can  be  manipulated  as  a  sub- 
stitute i'di-  the  function  —  thi-  may  be  said  to  be  the  uhimate  aim 
of  tiie  thcorv  ofdivei'gent   sei'ies. 

I'p  to  thf  |ir('sent  time  this  goal  has  been  reached  oidv  for  a 
re-ti'ieted  eja--  ot  (bvergeut    series.       J''urthei'more,  the  uniqueness 

i  St,.kcs,  !■•'■.  <«. 


di\i:r(;i:ni'  seimks  asd  co.ntinikd  fuacik >ns.     ii3 

of  correspoiulonce  iK'tween  the  function  and  the  series  has  been 
attained,  not  l)y  a  specification  of  tlie  i)roperties  of  the  function, 
but  by  means  of  some  algorithm  which,  wlien  applied  to.the  series, 
yields  a  single  fiuietion.  T'nquestionably  the  instrument  by  which 
the  greatest  progress  has  been  made  thus  far  is  the  integral.  The 
first  successes,  however,  were  reached  by  Ldc/uerre  *  and  Stieltjcs  f 
through  the  use  of  continued  fractions,  and  very  possibly  in  the 
end  the  continued  fraction  will  prove  to  be  the  best,  as  it  was  the 
earliest  tool.  J]ut  as  yet  it  has  been  applied  only  in  cases  in  which 
the  function  can  be  represented  under  the  form  of  an  integral  as 
well  as  of  a  continued  fraction,  although  ^\"ith  greater  ditticulty. 

To  explain  the  use  of  integrals  let  us  consider  the  familiar 
divergent  series  treated  by  Ijcjiicrrf, 

(1)  1  +,,.+  2!..^+:]!.-+.... 

This  is,  I  believe,  historically  the  iirst  divergent  series  from  which 
a  functional  cfjuivalent  was  derived.  ;};      Since 

*Sec'  No.  20  of  tlic  bibliography  at  llu'  end  of  lecture  t). 

i'  ]>il)liograj)hy,  No.  'ifia. 

i  Laguorre  [lo'-.  cit.)  gives  the  finietion  first  in  the  form  of  a  continued  fraction 
and  later  proves  its  identity  with  the  integral  which  gives  rise  to  the  divergent 
series.  IJorel  at  the  opening  of  the  second  chapter  of  Lch  Series  <litrri/rnl<'s  remarks 
that  '"  Laguerre  j)arait  avoir  le  premier  niontre  netteincnt  I'utilite  qu'il  pent  y 
avoir  a  transformer  une  serle  divergente  ...  en  une  fraction  continue  conver- 
gente.''  It  seems  almost  lo  have  escaped  notice  (see,  however,  p.  110  of  Prings- 
heim's  report,  Enei/kloiJuiUe  tier  Mntlt.  \\"i.<sPi,.<cJiitjten,  1  A  W),  that  Kuler  (IJiblio- 
graphy,  No.  4t)  j  derived  a  continued  fraction  fi-om  the  divergent  series 

1  --  lax  -)-  mi  iii  -'■-  n)j:'^  -\~  mini  -\  )t)  [m  -■-  '2ii)/^-\-  ■  ■  ■, 

uf  which  Laguerre's  series  is  a  special  case,  and  clearly  realizes  the  utility  of  the 
continued  fraction.  Moreover,  a  clos(>  i)ai-allel  to  the  course  followed  by  Lagiu-rre 
is  f(jund  in  the  woi'k  of  Laplace  who  derives  from  the  expression 

a  diveri^ent  series  and  from  this  in  turn  a  continued  fraction,  the  couvergents  of 
which  were  stated  by  liini  and  pi-oved  i)y  .Jacobi  to  ijc  alternately  greater  and  less 
than  the  expression.  Had  .lacobi  proved  also  the  convergence'  of  the  continued 
fraction,  the  work  of  Lagueri-e  would  have  had  an  exact  pai'allel  for  real  values 
of  f.      Cf.  Xo.  47  of  the  biblio-ratdiy. 


04  THE  BOSTON  COLLOQUIUM. 

/,*!=  r(,/,  +  1)=  r  ,-^-""(/., 

Jo 

the  series  may  be  written 

I     e-'dz  +  X  I     er---zdz  +  x-  1     t^'z^dz  +  ■  ■  ■, 

Jo  Jo  Jo 

the  path  of  integration  being  tlie  positive  real  axis.  If,  then,  by 
a  merely  formal  process,  the  sum  of  the  integrals  is  replaced  by 
the  integral  of  the  sum,  we  obtain 


r 


./o 

or  a  function 

(2)  /(..)=    rr-F(zx)dz, 

Jo 

in  which 

^■(-■)  =  y^«- 

The  function  thus  derived  is  an  improper  integral  which  has  a 
signilicance  for  all  values  of  x  exce})t  those  which  are  real  and 
positive.  It  can  be  shown  also  to  be  analytic  for  all  except  the 
excluded  values  of  .'■.  One  of  the  simplest  ])roofs  is  as  a  corol- 
lary of  the  filiowing  exce(Hlingly  fundamental  tlieorem  of  Willee- 
P<)iis.-<'ni/''  which  we  shall  have  occasion  to  use  again  later:  //'  in 
tlir  projjcr  lnfc(/r(d 

f{x,z)d- 


[ 


flic  iiif<yi-(nid  is  ('()i\ti inioua  in  z  (tnd  x  for  (ill  v<diieK  of  z  upon  the 
jKifli  of'  inli'f/ndion  (tnd  for  all  rdlncs  of  x  irilhin  a  rc(/ion  1  ;  if, 
Jiirtlit  riiKirc^  Jor  each  oj  the  oliorc  rallies!  of  z  it  /.s'  (inidjide  rn  x  over 
till'  rfijioii  T,  tlir  iiitff/rdl  irill  (dso  hr  an  (iniili/tic  function  if  x  i}i 
the  interior  of  T.  \\\  this  th(M)i-('m,  if  t  is  a  jioint  on  the  positive 
real  axis, 

r'c-dz 
1  1  -  '^•'• 

-*.\,n,.   <lr  l„  Snr.    Sriml.  ilr  Urn rrllr.<,    vol.    17   (ISD-J-:?),   ]).   :52;i. 


DIVERGENT  SERIES  AXD  COXTIXUED  FRACTIONS.       !>') 

will  represent  an  analytic  function  of  ,n  over  any  closed  ret^ion  of 
the  .'"-plane  which  excludes  the  })o.sitive  real  axi.s.  W,  now,  t 
passes  tlirough  any  indefinitely  increasing  set  of  values, 

K  <  U,  <  ^3>  •  •  •> 

we  have  in 

./.(■'• 


■'^I'T-t. 


a  series  of  analvtic  functions  which  is  seen  at  once  to  convertre 
uniformly  over  the  reu'ion  considered,  since 


for  sufficiently  great  values  of  /  and  j.  The  limit  (2)  is  therefore 
analytic. 

By  deforming  the  path  of  integration  the  same  conclusion  con- 
cerning tlie  analytic  character  of 

the  function  (2)  can  be  extended    ■ '        i >. 

^-   ■'  _       o  X  ^ 

to  all  values  of  x  upon  the  posi- 
tive real  axis  excepting  0  and   oc,  and  when   the  deformation  is 
made  on  o])})osite  sides  of  a   fixed  point  .'■,  the  two  values  of  the 
integral  will  be  found  to  differ  by 

1       1 

(3)  2!7r~e~-. 

The  integral  accordingly  rej)resents  a  multiple-valued  function 
with  the  singular  points  0  and  x,  the  various  branches  of  which 
differ  from  one  another  by  multiples  of  the  period  (o).  For  the 
initial  branch  which  was  given  in  (2)  the  limit  of /"(.'■)///!  will 
be  the  (//  +  l)th  coefficient  of  (1)  if  .'■  ap})roaches  the  origin 
along  any  rectilinear  path  excej)t  the  positive  real  axis. 

Let  the  process  which   has  been  adopted  for  the  series  of  Iji- 
guem  be  applied  next  to  any  other  series 

(I)  ",,  +  ^'i.''  +  «■,■>'-  -\ 


!'G  TI[E  BOSTON  COJ^LOQUIUM. 

having  a  finite  radius  of  convergence.      If  we  write  the  series  in 
the  form 


"„  +  ",•'■  + ^M  -)'.  ■-  +  •••  +  "!  ,;;  ■^"  +  ---, 


then  rej)hu'e  the  factor  u  I  by  its  expression  as  a  F-integral,  and 
finally,  l)y  a  stej)  having  in  general  only  formal  significance,  bring 
all  the  terms  under  a  coimnon  integral  sign,  we  shall  obtain 

(4)  J  e-F(z,yh, 

in  which 


This  integral  is  the  expression  upon  which  Borrl  builds  his  theory 
of  divergxMit  series,  and  may  be  regarded  as  a  generalization  of  a 
very  interesting  theorem  of  Cae.sarn/''-  The  series  (5)  is  called 
the  (/ss()i-iiifc(I  ,sY'/vV.N'  of  (I). 

Two  cases  are  now  to  be  distinguished  accoixliiig  as  the  funda- 
mental series  (i)  has,  or  has  not,  a  radius  of  convergence  Ii  which 
is  gi'cater  than  0.  If  the  radius  is  not  zero,  the  associated  series 
has  an  infinite  radius  since 

lini  *  '        =  Inn   \|  =  'N 

„    ,.    N  "  •       „    /,     >  "  • 

ami  it  accordinglv  represents  an  cntii'c  function.  It  is  a  simjtle 
matlci'  tn  pi'ovc  that  the  integral  (  1)  will  have  a  sense  if  ./•  lies 
within  llic  cii'clc  of  convci'gcncc  (»f  ( I  ),  and  that  the  values  of  the 
intcgi'al  and  scric-  arc  identical.  l)Ut  the  integral  niav  al>o  have 
a  -eii-e  fill' value-  of"  .'■  which  lie  without  t  he  circle,  and  in  thisca>e 
the    inte;^i-al    mav  be  ii<ed    to  net    t  he  aiiah't  ic  cout  inu;U  ion    of  (  I  ). 

•    (     I.     l',.,lrl,     /.,,      S/,  /.  -    ■hr.  ,-/.„^.-.     p|..     ^S     '.In. 


I)IVER(a:XT  SKRIES  AND  COXTrXlKD  FRACTIONS.       !'< 

The  series  is  said  hv  llorcl  to  l)e  xviDnuiftlc  *  at  a  jmiiit  ,'•  when  the 
integral  (4)  lias  a  meaning  at  this  jxjint. 

The  second  case  is  that  in  which  the  fundamental  series  is 
divergent.  The  associated  series  in  this  ease  n^ay  be  either  con- 
vergent or  divergent.  If  it  is  convei'gent  oidy  over  a  j)orti()n  of 
the  j)lane  of  a  =  ':.>',  we  are  to  understand  hy  F(ii)  not  mei'ely  the 
value  of  the  associated  series  l)ut  of  its  analytic  continuation. 
Let  X  f)r  an  instant  be  given  a  iixed  value.  Then  when  z 
describes  the  ])ositive  real  axis,  v  in  its  plane  describes  the  ray 
from  the  origin  jxissing  through  the  point  x.  l^  l'\ii)  is  holonior- 
|)hic  along  this  ray,  it  is  possible  that  the  integral  (4)  -will  have  a 
sense.  Su])pose  that  this  holds  good  as  long  as  x  lies  within  a 
certain  sj)ecified  region  of  its  pla.ne.  Then  for  this  region  a  func- 
tion will  be  ol)tained  uniquely  from  the  divergent  series  by  the  use 
of  the  integral,  j^recisely  as  in  the  case  of  the  series  of  Jjif/uerre. 

This  method  of  treatment  is  obviously  restricted  to  divergent 
series  for  which  the  associated  series  are  convergent,  and  it  will 
not  always  be  a])plicable  even  to  these.  A  divergent  series  in  which 
there  is  an  infinite  number  of  coefficients  of  the  same  order  of  mag- 
nitude as  the  corresponding  coefficients  of 

(6)  1  +  :/;  +  (2  Vfx-  +  (;]  !)-.';^^  -^  ...  +  (n  !)V  -f  •  •  • 

can  not  l)e  summed  in  this  maimer.  It  will  be  noticed,  however, 
that  the  series  just  given  is  one  whose  lirst  associated  series  is  the 
series  of  lAjr/iwrrc,  and  whose  second  associated  series  is  conse- 
(picntly  convergent. 

The  method  of  Bore/  can  be  readily  extended  so  as  to  take 
account  also  of  such  series,  or,  more  generally,  of  series  that  have 
an  associated  series  of  the  itth  order  wiiich  is  convergent.  (3ne 
mode  of  doing  this  is  l)y  the  introduction  of  an  »-fold  integral. 
Sui)pose,  for  examjile,  that  in  (G)  one  of  the  two  factorials  /i !  is 
replaced  by 


f. 


*  Some  otlier  term  would  be  i>refer;il)le  since  his  definition  refers  only  to  one 
of  many  possible  modes  of  suinmation.  A  series  may  be  simultaneously  "  sum- 
mable"  at  a  point  .'■  by  one  method,  and  non-summable  by  another. 


^^  THl-:   lU)STOX  COLLCXa'TUM. 

and  the  other  l)v 

tJn 

The  (//  -f  l)th  term  of  tlie  series  becomes 

X"  I      I     e~'-~-:c'H"(1z<Jt, 

Jo     Jo 

and  \ve  obtain  the  two-fold   intes:ral 

(hdf 


Jo     X      1    -  f^^' 


for  the  i'linctional  e(|uivalent  of  tlie  series.  This  is  a  function, 
the  initial  l)ran('h  of  which  is  analytic  over  the  entire  plane  of  r 
excej)t  at  the  ])oints  0  and  dc. 

We  turn  now  to  the  consideration  of  the  reg;ion  of  snmmabilitv, 
in  wliicii  .'•  must  lie  in  order  that  the  integral  siiall  have  a  sense. 
/>o/>7  has  determined  the  shai)e  of  this  region  when  the  funda- 
mental series  {\)  is  convergent,  but  in  so  doing  he  restricts  him- 
self to  what  he  calls  the  (ih.^ohifclji  suniuKr/jfc  series.  The  series 
(I)  is  said  to  be  (ih.^olufclji  .•oi/tuitdhff  for  any  value  of  .'•  when  the 
integral  (1)  is  al>solutely  convergent  and  when,  furthermore,  the 
successive  integrals 

(")  ^  ,,...      ''-^  (X=  1,2,  ...) 

have  also  a  sense. '^^ 

To  tix  the  shape  of  the  region  />o/7'/  shows  first  that  il'  a  lune- 
tiou  (Ictined  l)y  a  (;onvei'gent  series  (  I )  is  absolutely  summablc  at 
a  |»oint  /\  it  is  analytic  within  the  circle  described  upon  th(!  line 
O/'  as  diametci',  connecting  /'  with  the  origin  O ;  convei'selv,  if  it 
is  analvtic  within  and  Uj)on  a  cii'cie  having  ^)/'  as  diameter,  it 
mu^^t    be   al)>obitcly   summai)lc   along  0/\  inclusive  of  the    point 

* 'I'lu'  ci'iidition  7  I  \v;i-  not  (iri;^in:ill_v  incliidcd  in  liorel's  dflliiil  idii  uf  ulinK- 
Intf  i^iiiiiinniiililv  i  Aim.  il>-  I' J:'/-.  .\nr.,  -er.  :'>,  vol.  Kl,  1>'.»'.M,  .'ni(i  is  -upiTtluoiis 
ill  tixiii;,'  tiic  -lia|.c  (.!'  the  n,t;i(m.  (  f.  M.ith.  Ann.,  V(.i.  .")•')  ll'.i(l2),  ]..  74.  Tlie 
iiKidilicjiinii  of  the  dflinilioii  \\:is  iiit  roduceii  in  tin-  Si'ri'.--  iiiirr'/>  nifs  iiiid  is 
iiccdfij  t'oi-  I  he  dc\  I'lopinciit-  cxjilaincd  ludow,  p.  ll)"_*.  <  haiitcfs  ;>  :ind  I  of  this 
treati-f  can  lie  read  in  ronncctioii  with  liic  i^rt'sent  Iccturf. 


i)iver(jp:xt  series  and  continued  fractions.     r>j) 

P .  As  /-*  moves  outward  from  the  origin  along  any  rav,  the  lim- 
iting position  for  the  circle  is  one  in  which  it  first  passes  through 
a  singular  ])oint  S,  and  at  this  point  8P  and  08  subtend  a  right 
angle.  The  region  of  absolute  summability  can  therefore  be 
obtained  as  follows  :  Mark  on  each  ray  from  the  origin  the 
nearest  singular  point  of  the  function  defined  by  (I),  if  there 
is  such  a  ])oint  in  the  finite  plane.  Then  through  this  })oint 
draw  a  perpendicular  to  the  line.  Some  or  all  of  these  per- 
pendiculars will  bound  a  polygon,  the  interior  of  which  con- 
tains the  origin  and  is  not  penetrated  by  any  one  of  the  perpen- 
diculars. This  region  is  called  the  polyrjon  of  suvimdbiliti/.  If 
the  singularities  of  the  function  are  a  set  of  isolated  points,  the 
polygon  will  be  rectilinear.  For  the  extreme  case  in  which  the 
circle  of  convergence  is  a  natural  boundary,  the  polygon  and 
circle  coincide.  In  every  other  case  the  circle  is  included  in  the 
j)olygon.  '^rhus  by  the  use  of  (4)  Bore/  effects  an  analytic  con- 
tinuation of  the  series  over  a  })erfectly  definite  region  whenever  an 
analytic  continuation  exists.  On  passing  to  the  exterior  of  the 
polygon  the  series  ceases  to  be  absolutely  summal)le.  As  an 
cxam])le  of  this  result,  take  the  series 

which  is  the  familiar  ex])ansion  of  ^  log  (1  +  ''")/(^  —  •*")•  The 
lingular  points  of  the  function  arc  -f  1  and  —  1,  the  circle  of 
<'()nvergcncc  is  the  unit  circle,  and  the  jiolygon  of  summability 
is  a  strip  of  the  ])lane  included  between  two  perpendiculars  to  the 
real  axis  through  the  points  =i=  1. 

When  the  given  series  is  divergent,  the  form  of  the  domain  of 
sammai)ility  has  not  been  determined  with  such  precision.  The 
only  information  which  wc  have  upon  the  subject  is  contained  in 
a  brief  but  important  comnuinication  by  Phraf/tnea  in  the  ( 'ornp- 
fcs  Roidiis/'-  published  since  the  appearance  of  BorePx  work. 
Ph.r(i(/inen  considers  here  the  domain,  not  of  absolute,  but  of  sim- 
j)le  summability  for  JMpla<-e\H  integral 

•■Vol.  132,  p.  139();  June,  1901. 


100  THE  HosTox  ('OLi.o(n^ir^r. 

(8)  r  e-^f(z,'yz, 

in  which _/'(2:x)  denotes  an  arbitrary  function. 

To  adopt  a  term  of  3fitfa(/--LeJffer,  tlic  domain  is  a  "star," 
wliich  is  derived  as  follows  :  Draw  any  ray  from  the  origin.  If 
the  series  is  summable  at  a  ])oint  i*',,  of  this  line,  Plu-arpnen  sliows 
that  it  is  summable  at  every  ])oint  between  x^,  and  the  origin  0. 
There  is  therefore  some  point  P  of  the  line  which  sej^arates  the 
interval  of  summability  from  the  interval  of  non-snmmability. 
If  the  function  is  summable  for  the  entire  extent  of  the  ray,  P 
lies  at  infinity.  In  any  case  let  the  segment  OP  be  obliterated 
and  then  make  a  cut  along  the  remainder  of  the  line.  When  the 
same  thing  is  done  for  every  ray  which  terminates  at  the  origin, 
there  is  left  a  region  called  a  star,  bounded  by  a  set  of  lines  radi- 
ating from  a  common  center,  the  })oint  at  infinity. 

P/rr(tf/inen  says  that  the  ])roof  of  this  result  is  so  sim])le  that  it 
can  be  given  "  en  dcnx  7/(o/.s'."  For  this  reason  1  shall  rejiro- 
duce  it  here.  AVe  are  to  show  that  if  the  integral  converges  for 
anv  value  x  =  x,,,  it  will  also  converoe  for  x  =  6x„,  if  0  <  ^  <  1. 
Place 

Jlzx^)  =  ct>(z)  +  if{z). 

For  .('  =  .r^  the  real  and  imaginary  com])onents  of  the  integi'ds, 

(9)  r^{^y-<h,         ;f^^(z)r-.Jz, 
have  a  sense.      We  are  to  ])rove  that  the  integrals 

(10)  r  (jyizOy-  <lz,       I    f{zOy-<h:, 

obtaiiH'd  bv  I'cplaciiig  .>\^  bv  Oj\^,  also  exist.      ('onsidci'  cither  inte- 
gral, for  exaiii])I('  the  formei'.       Let  (•.^^f/^  -^  "., --^  ^ ,  JUid  put 


DIVERGENT  SERIES  AND  CONTINUED  ERACTIONS.    101 

Jiy  the  c!uui<^c  of  variable  '/•  =  dz  this  becomes 

Since  r""  '  **"''  is  a  positive  and  decreasing-  function  in  the  interval 
considered,  the  second  mean-value  theorem  of  the  integral  calculus* 

may  l)e  a])j)lied,  giving 

r7,a-i9i   pea 
(11)  ./=        ^         I      cf)(>r)e-"(Jir, 

in  which  a  designates  an  apjiropriate  value  between  a^  and  a^. 
This,  as  Fhr(i(/)iien  says,  proves  the  theorem,  but  a  word  or  two 
of  explanation  additional  to  his  "  deux  tnof.i "  may  not  be  unac- 
ceptable to  some  of  my  hearers.  The  necessary  and  sufficient 
condition  for  the  existence  of  the  first  of  the  two  integrals  given 
in  (10)  is  that  by  taking  two  values  «,  and  a.,  sufficiently  small  or 
two  values  sufficiently  large,  the  integral  J  may  be  made  as  small 
as  we  choose.      Now  this  is  true  of 


f 


(^('rj(.'    "(Ill- 


since  the  integrals  (9)  exist,  and  ecjuation  (11)  show  then  that  it 
must  be  true  likewise  of  J  because  the  factor  (-"^('^-^> jO  has  an 
upper  limit  for  0  <  ^^  <^  ^  <;  1  and  0<«j<oc.  It  follows 
therefore  that  the  integrals  (10)  exist. 

Two  other  facts  stated  by  Plirdgiacn  are  also  of  interest.  The 
function  of ./;  defined  by  (8)  i.s  a  monogenic  function  which  is  holo- 
morphic  at  every  jioint  in  the  interior  of  a  circle  described  upon 
OP  as  diameter,  li,  also,  in  place  of /(.t.r)  we  take  the  associated 
series  F(zx)  of  a  convergent  series  (I),  the  star  of  convergence 
coincides  with  BnreP.^  polygon  of  absolute  sunimai)ility.  Thus 
the  regions  of  absolute  and  non-absolute  summability  are  the 
same,  av  differ  at  most  only  in  resi)e('t  to  the  nature  of  the  boun- 
dary })oints. 

*  I>oniiet's  form  :  Enrj/LlojHii/ie  dcr  Math.    H7xy. .  11  A  2,  i<  ;>5. 


1<»-!  THE  r>()ST()N  COlJ.fX^UJUM. 

It  might  1)0  thought  tluit  the  result  of  PJn-dfjmcn  makes  the  con- 
cej)t  of"  absolute  summahility  useless.  This  is,  however,  in  no 
wise  the  case.  At  any  rate,  Ilorcl  employs  the  concept  to  estal)- 
lish  the  important  conclusion  that  a  divergent  series,  if  absolutely 
sunnnal)le,  can  be  manipulated  precisely  as  a  convergent  series. 
Thus  if  two  absolutely  summable  series,  whether  convergent  or 
divergent,  are  multi])lied  together,  the  resultant  series  will  also  be 
al)solutelv  summable,  and  the  function  which  it  defines  will  be 
the  sum  or  product  of  the  functions  defined  by  the  two  former 
series.  Or,  again,  if  an  absolutely  summable  series  is  differen- 
liated  term  by  term,  another  such  series  is  obtained,  and  the  latter 
yields  a  function  which  is  the  derivative  of  the  one  di'fined  by 
the  former  series.  Lastly,  the  function  determined  by  an  abso- 
lutely sunnnable  series  can  not  be  identically  zero,  unless  all  the 
coefficients  of  the  series  vanish. 

These  facts  make  j)ossible  the  immediate  application  of  BorePs 
theory  to  differential  e<piations.      \^,  in  short, 

P{.>',ii,!i',---,ir^)  =  ^ 

is  a  differential  e(|uation  which  is  holomorphic  in  ;r  at  the  origin  and 
is  algebraic  in  //  and  its  derivatives,  any  absolutely  summal)le 
s(M'ies  (I),  wliich  satisfies  formally  the  e(juation,  defines  an  analytic 
function  that  is  a  solution  of  the  equation.  For  examj)le,  it  will  be 
f  )und  that   the  series  of  LiKjncfi-c  satisfies  formally  the  equation 

X-  :  +  (.'■-  i).v==-  1 

and  hence  the  function 


■'■'■■■- i, 


I  —  ,^.(• 

nui<t  be  a  solution  of  the  ('(luatioii. 

TIm'sc  conclusions  of  Ilnrr/  should  be  sti'oiiglv  emphasized. 
Ill  aiiv  complete  tlu'orv  of  divergent  series  it  is  an  ultimatum 
that    tliev   shall    in   all    essential    points  '^  |)ermit    of"   manij)ulation 

■  In  ;ui  al)~iiliitcly  suinninlilc  -cries  it  i-  iiui  always  Ic^^'it iinate  to  rlianj^'c  the 
onieriif  :iii  inrmltc  iiuinlifr  nf  tt-i'in^.  <  t'.  \'.i>iv],  ,J"urii.  <li  M'llli.,  st-r.  •">,  vol.  2 
(  I^'.h;).  ,,,   111. 


I)1Vkij(;i:nt  skkiks  and  coxtimfj)  fkactions.    lo". 

procisclv  as  converoent  series,  this  ])ro])erty  beinn^  a  recjuisite  for 
satisfactorv  a])i)licati()n  to  diifereiitial  equations. 

In  oiir  ])reee(ling  exposition  of  Bore/'s  theory,  we  have  intro- 
<hu'e(l  his  ehief  integral  hv  a  method  whicli  j)ermits  of"  exj)ansion 
in  varions  direetions.  L<  Uoii  in  Ids  very  excellent  thesis*  snggests 
a  change  of  the  function  in  L(ij)f(KU''.<  integral  which  greatlv  en- 
larges the  aj)plical)ility  t  of  JinreC)^  metiiod  without  essentially 
changing  its  character.      Jvct   the  initial   series  (I)  he  first  written 

". + -~ni'  + 1)  p(^;'+ ,) + "Mi,  + 1)  p^.;;^  ,j  +  ■  ■  ■ 

and  then  replace  the  second  factor  in  each  term  hy 

J'  t/O 

This  gives  for  the  formal  e(|uivalent  of  the  series  the  integral 
in  \vhich  the  associated  function  is  now 

The  numhcr  j)  r(!mains  to  he  fixed.  If  the  series  (I)  is  divergent, 
there  is  a  critical  value  of  y>  such  that  any  smaller  value  of  y> 
gives  an  associated  series  having  a  zero  radius  of  convergence, 
while  a  lai'ger  value  gives  one  with  an  infinite  radius  of  conver- 
gence.     This  critical   value  [/  mav   he  said   to  gaug(>  or   measure 

■^  A,ninlr.-<<l€  Toulon.^,',  ser.  2,  vol.  2  (l'.U)O),  p.  -lli;. 

i  Since  tliis  was  written,  n  very  interestiui;;  applieation  of  \.e  Itoy's  idea  to 
(lifTerential  eipiations  has  lieen  made  liy  .Ntaillet,  Ann.  de  F  K<-.  X')!-.,  ser.  '■>,  \-ol. 
20  (IS!).-,),  p.  4S7  fi; 


1<*1  THE  BOSTON  COLLOQI'IU^r. 

the  (lei^rec  of  divergence  of  tlie  series.  For  the  divergent  series 
treated  hy  Ilorel,  p'  =\.  If  y/  =  0,  the  series  (I)  has  a  finite 
radius  of  convergence.  On  the  other  liand,  \vlicn  ;/  =  oc,  Ix  Roifn 
integral  can  not  be  applied,  but  it  may  be  conjectured  that  such 
cases  will  be  of  very  rare  occurrence.  Ij'  Roi)  proposes  to  employ 
the  integral  when  the  associated  series  is  convergent  for  p  =  p'  and 
when  also  its  circle  of  convergence  has  a  finite  radius  and  is 
not  a  natural  boundary.  ^J'he  function  obtained  from  (12)  will 
be  unique,  and  he  shows  that  the  series  which  are  suramable  by 
its  use  like  the  series  of  llorel,  can  be  manipulated  as  convergent 
scries.  One  might  also  incpiire  wiiether,  in  case  (l.">)  diverges  for 
p  =  p'  and  we  take  /)  >y/,  we  shall  not  get  a  unique  result  irre- 
spective of  the  value  of  jj. 

Other  forms  of  integrals   may  also  be  selected  for  the  summa- 
tion of  the  series,  as  for  example,''' 


£ 


in  which 

F{zx)  =  /5„  +  /3.^.r  +  l^/x'  +  •  .  • . 

To  generate  the  given  series  (I)  we  must  so  select/  (.r)  and  F(zx) 
that 


'.,  =  /^„  /     ./1-)  ■  ^"''^- 


Jlorrf  chooses  for  _/"(ij  the  exj)onential  iunction,  making  in  conse- 
(|U('nce  /■'("..''),  his  associated  series,  de[)endent  only  ui)on  the  given 
sci-ies.  Hence  his  jiroccss  is  calked  verv  appropriately  the  ex- 
|)oncntial  method  of  siiuimat ion.  N/Zr/Z/Vx,  r  on  tlie  other  hand, 
with  his  continued  fraction  ai'rivcs  at  an  integral  in  which  /'(")  is 
thi'  tixcd  fiMictioii  and  /(■- )  i<  the  vai'iable  function  dependent  on 
th(i  series  Lciven.       ^^)r  the  fixed  Junction  he  takes 


I   —  '..'■ 

CL     I,r    l:.,_V,    A..-,    rr.,    [,],.      II    1  -11.",. 
L'.r.    rit. 


DIVERGEXT  SERIES  AND  COXTIXUED  FRACTIONS.     lO") 

so  that 

(14)  a,  =    f  J\z).z"dz. 

Jo 

At  first  si^ht  this  choice  of  functions  wonld  seem  to  be  a  very  desir- 
able one,  for  the  function  defined  by  the  diverg-ent  series  is  ol)tained 
in  the  familiar  form 


(15) 


«=.-,fet 


Upon  examination,  however,  it  turns  out  to  be  otherwise.  For 
suppose  the  divergent  series  to  be  given  and  f{z)  is  to  be  found. 
The  pro])lem  is  then  a  very  difficult  one,  that  of  the  inversion  of 
the  integral  (14)  when  a^^  is  given  for  all  values  of  u.  This  is 
what  Slielfjes  terms  "  the  problem  of  the  moments."  It  does  not 
admit  of  a  unique  sohition,  for  Stielfjes  himself*  gives  a  function, 

f(z)  =  c~'^'-  sin  V  ,^, 

which  will  make  ((^=  0  fi)r  all  values  of  n.  If  tiie  sup})lementary 
condition  is  im])osed  that  j\z)  shall  not  be  negative  between  the 
limits  of  integration,  only  a  single  solution  J\z)  is  })ossible,  but 
the  divergent  series  is  thereby  restricted  to  belong  to  that  class 
which  Sfirltjes  derives  naturally  and  elegantly  l)y  the  considera- 
tion of  his  continued  fraction. 

Thus  far  our  attention  has  been  confined  exclusively  to  integrals 
in  which  one  of  the  limits  of  integration  is  infinite.  There  are, 
however,  advantages  in  using  a])propriate  integrals  having  both 
limits  iinite,  at  least  if  the  given  series  is  convergent  and  the 
integral  is  used  for  the  purpose  of  analytic  continuation.  In 
])arti('ular,  the  integral 

(IG)  /(.,')  ^  f  V(z)F(z,^yJz 

Jo 

should  be  noted,  to  which  IFddaiiuirJ  has  drawn  attention  in  his 
thesis. t      This   fiUls  under  Vdlh'c-PonHxi n^s  theorem  when    V{:c)  is 

*L<,r.    cit..    i   oo. 

jJouin.  ill-  M'tth.,  ser.  4,  vol.  S  (1S'.)2),  [.p.   loS-lCo. 


lOG  l-HK   I'.OSTON  COLLOQUIUM. 

oontimions  alonu"  the  ]xitli  of  into<^ration  and  when  also  J"\(()  is 
analytic  in  n  =  xx  for  all  values  of?;  upon  the  path  of  integration 
and  for  vahies  of  .»•  in  some  sjiecitied  region  of  tlie  .r-])lane.  If, 
as  we  snp])ose,  the  path  is  rectilinear,  the  values  of  x  to  be  ex- 
cluded are  evidently  those  which  lie  on  the  ])rolongations  of  the 
vectors  from  the  origin  to  the  singular  })()ints  of  F(:ry  The 
region  of  convergence  of  (10)  is  consequently  a  star,  whose  ])ouu- 
dary  consists  of  j)rolongation  of  these  ve(;tors.*  Tluis  IhuhiiimnVH 
integral,  when  applied  to  the  analytic  continuation  of  a  function, 
is  suj>erior  to  Borers  in  the  extent  of  its  "region  of  suniniability." 
This  is  ilhistrated  in  Lc  //oz/'x  thesis  t  with  tl)e  \oy\  familiar  series  : 


1    /•'       -'V/- 

^.1    I   -iX  -  z] 


Here  the  coetTicient  of  ./"  is 

I    ^( 
so  that 

1  r         '^2 

-rrj.     ,    ,(l_.)(l_.,r) 

Since  /•'(--'')  =  1/(1  —  "i-'O,  the  I'egion  of  sununahility  is  the  entire 
j)lane  oi"  .r  \vitii  the  exception  of  the  ])art  of  tlie  real  axis  hetween 
.r  =  1  and  ./■  =  or  .  //ry/v/'x  pojvgon  of  siniunal)ilitv  for  the  series, 
on  the  other  hand,  is  onlv  the  half  plane  lying  to  the  left  of  a 
j)ei-|»endienlar  to  the  I'cal  axis  thi'ough  the  ])oint  .'•  =  1. 

Mneh,  it  seems  to  me,  can  vet  l)e  dou(>  in  I'ollowing  u]»  the  use 
of  llndd iiiif rd'a  iiitegi'al.  (  )iie  sjx'cial  case  has  heen  studied  alreadv 
l>y   /,'    //"_;/,  ill  which  the  (n   f   1  )ili  coenicient  of  (I)  has  the  form 


^> 


^(■.)'l:. 


I'lii-  <■'  ■mlii-idii  iilr^ii  luilcU  it'  only  /      1 '(  •.  "/•.  is  an  ali^dliitclv  coinrr-uciit  intc 
fjra  i.  a-  i-  -In  i\\  n  hv   1  laijaiiia  ii|. 
i  :■.   II  1. 


i)i\i;K(ii:N'r  skrtks  and  contimI':])  fkactioxs.    i<>7 

The  series  therefore  deHiies  ii  function 

X  1  -  ^-^ 

whicli  is  analytic  over  the  entire  ])lane  except  along  tiie  real  axis 
between  .'.■  =  1  and  .c  —  qc,  '^i'he  j)ath  of  intej^nition  may  also 
permit  of  deformation  so  as  to  show  that  the  cut  between  the 
points  is  not  an  essential  cut.  It  is  interesting  to  note  that  if 
0(2:)  is  ])ositive  between  0  and  1,  the  primary  branch  of  the  func- 
tion has  only  real  roots  which  are,  moreover,  greater  than  1.* 

LKcTrKK  .').  On  flic  Dticrmi it((fioii  of  tlic  Sinf/n/arifie.s  of  Func- 
tions Defined  hy  Pover  .SV/vV.s'. 

Up  to  the  })resent  time  comparatively  little  successful  work  has 
been  tlone  in  determining  the  singularities  of  functions  defined  by 
power  series,  and  the  little  which  has  been  done  relates  mostly  to 
singularities  u])on  the  circle  of  conv(M-gence.  AVork  of  this  special 
nature  1  shall  omit  from  consideration  here,  thus  passing  over  the 
memoirs  of  Fabry,  and  I  shall  call  your  attention  to  the  literature 
which  treats  of  the  singularities  in  a  wider  domain. 

The  most  fundamental  and  practical  result  yet  obtained  is 
undoubtedly  a  brilliant  theorem  of  ILuhunard,^  in  the  wake  of 
which  a  number  of  other  interesting  memoirs  have  followed. 
This  theorem  is  as  follows  : 

//'  liro  annljitic  functions  arc  defined  by  tJw  conver(/ent  poicer  .•<e/'ien 

(1)  cf){x)  =  a^  +  a^x  +  a^r  +  •  ■  • , 

(2)  ^(,r)  =  b^  +  b,,  +  b^.-'+..., 
the  only  singularities  (f  tlie  function 

('')  ./'(•'■)  =  ^',/>o  +  ^'i^^r'"  +  ^'2^^2-«'  +  •  •  • 

laill  be  jtoinfs  u'/am'  a_tflxes  7..  cr/v  tlic product  <f  aljixes  of  the  singu- 

t(ir  j)oinfs  a.  and  f^.  (f  the  frst  tu-o  functions. 

■■  Lv  Koy,  h,r.  cil..  pp.  ,">:^(i-:!:'>l. 

t  Arin  Mall,.,  voi.  L'l'  (  IS'.tS),  ]..  .V.. 


1*^8  THE  BOSTON  COLLOQUIUM. 

The  p(3ssibility  that  x  =  0  should,  in  addition,  be  a  singular 
j)oint  has  been  pointed  out  since  by  L'nide/of. 

Although  Jl(i(l((iii((r(P.s  proof  of  the  theorem  is  not  a  compli- 
cated one,  I  shall  })resent  here  a  still  simpler  })roof  given  by  Bore/.* 
liCt  7i*  and  /*'  be  tiie  radii  of  convergence  of  (1)  and  (2)  respec- 
tively, and  take  a  number  p  such  that  iiV/3  !>  1  /'  •  If  then 
\''X\^\  p.r\  <^  Ji  and  |.7'[>17^',  the  product  of  </)(-.>■)  and 
ylr(\  jj')  can  be  devel()])ed  into  a  LdurenVf^  power  series  which  is 
valid  in  a  circular  ring  in  the  a;-j)lane,  having  its  (;cnter  at  the 
origin  and  the  outer  and  inner  radii  Ii  p  and  1  //'  res])ectively. 
In  this  ))nKhict  the  absolute  term  is  obviously 

(4)  /(-^)  =  <',/%  +  "J>:-  +  <'J'^  4-  •  •  •  . 

Consider  now  the  integral 


('■>) 


'liir , 


in  which  r  is  a  closed  path  surrounding  the  origin  and  contained 
within  the  circular  ring.  As  long  as  ':  in  its  plane  lies  within  a 
circle  of  radius  p  <^  lUi' ,  having  its  center  in  the  origin,  the 
integral  will  surelv  define  a  function  of  z,  and  this  function  is 
evidentlv  e(jual  to  the  residue  of  the  integrand  for  .'"  =  0,  which 
is  J{z). 

We  ^hall  now  seek  to  extend  this  function  by  varying  :.  and  at 
the  same  time  (leformingappi"oi)riatel v  the  j)atli  of  integration.  l>y 
the  tlicoi'cm  of  \'(illrr  /*iiiissiii  (|Uot(Ml  ill  Lecture  !2,  the  integral 
will  continue  to  re])reseiit  an  analvtic  function  of  z,  provided  at 
evcrv  -lagc  the  integrand  i-cmaius  analytic  in. /•and  z  ;  .v  being 
an\'  |ioiiit  upon  the  path  of  integration.  Now  the  valines  to  be 
avoided  are  clearlv  tlie  singular  j)oints  of  the  functions  (f){z.r)  and 
■yp-i  1  /.'•  I  ;    naniciv  the  ])oints  : 

*    llnll,    ,/.    I,,   Snr.     Mall,.    ,/,     Fpnir,,    v,,l.   -Jl')    I   IS'JS),    pp.    •JliS-'J-lS. 

An  iiitiTc-~l  iiiLT  prodI'    '•(■<(  iiiiiUi  (■.!.<:•"   is  ^ivcn  witlimil    the   use  of   iiitcrjrals   liy 
I'ill.  l,,rl,.   in   Ihr   l!,n.i;rn„ln  .I.Un    II.  .l-r,/,/.  J.lh  Sri,„-.,   ,/,    llnhupm.   new  srr.,    vol. 


I'l' 


i)ivkk(;i:nt  seihes  and  ('ONTixuki)  fkactions.   109 


_  _  ^ 

—  .,    .'.  —  ^. 

The  points  ,'•  =  1  /?.  lie  A\itliin  the  circle  (1  //')  \vhieh  is  the  inner 
circumference  of  tlie  ring',  while  the  points  x  =  a.'y:  before  the 
variation  of  :■:  lie  without  the  outer  circumference  (I*ip).  For 
sim])licity  of  presentation  it  may  he  convenient  to  assume  at  first 
that  tiiese  jioints  form  an  au;t2:rec;:ate 
of  isolated  points.  Su]>pose  then 
that  z  follows  any  jiath  in  its  })lane 
emeririiig  from  the  circle  (p).  Then 
the  points  cc.z  describe  certain  coi'- 
responding  patlis  which  we  will 
mark  in  the  .i'-])lane.  At  the  same 
time  th(^  contour  c  mav  l)e  deformed 
continuouslv  so  as  to  recede  before 
the  ])oints  a.'z  without  sweeping 
over  any  point  1  '/3.,  ))rovided  merely  that  a.'z  never  collides  with 
a  point  1  73. ;  that  is,  z  must  never  pass  through  a  j)oint  a./3.. 
Now  when  z  is  held  fixed,  a  deformation  in  the  contour  c,  sid)ject 
of  course  to  the  condition  indicated,  produces  no  change  in  tlie  value 
of  the  integral  /(z),  since  the  integrand  is  holomorpln'c  between 
the  initial  and  deformed  paths.  On  tlie  other  hand,  when  the 
path  is  k(>])t  fixed  and  z  is  varied,  we  have  tlie  analytic  continu- 
ation of  _/"(,i)  in  accordance  with  the  theorem  of  ValUe  J/'oiis><h>. 
By  the  two  changes  togetlier_/[rc;)  maybe  continued  over  the  entire 
plane  of  z  with  the  exception  of  the  points  a./3.  =  7. .  To  these 
sliould,  of  course,  be  added  z  =  oc,  also  ;s  ==  0  as  a  possible  singular 
point  for  any  branch  of/(z)  except  the  initial  brancli. 

It  should  be  observed  that  7..  is  shown  to  be  a  potential  rather 
than  an  actual  singular  point.  When,  however,  it  is  such  a  j)oint, 
the  cliaracter  of  the  ])oint  dejiends  in  general  solely  ujion  the  nature 
of  the  singularities  m..  and  /3.  for  (1)  and  (2)  respectively.  This  fact 
was  noticed  by  Bore/  and  demonstrated  in  the  following  manner. 
Tet 


110  THK  BOSTON  ('()LL(  XjriU.Ar. 

l>t'  any  convoru-ent  series  defininjx  a  I'liiieticMi  <^|(.'')  wliieli  is  regu- 
lar at  a .  Then  (j)^{x)  =  (f)^(?')  +  (f)(.r)  is  a  function  which  lias  at 
y..  tlie  same  singularity  as  (p{:>').  The  combination  of  the  series 
foi-  (^-A-'')  'I'l*^  ^"•*^'  i^(:'')  ^^y  itoiUiiiuird'H  process  gives  the  function 

in  wliicli 

Xow  since  ^,(.'')  is  regular  at  oc.,  wlien  compounded  with  -yjrh'^ 
it  must  give  a  function  ./',(.'•)  which  is  regular  at  7,  .  It  follows 
that  /..{.'■)  an<l  ./'(.'•)  have  the  same  singularity  at  y...  Thus  the 
nature  ot'  this  singula]'  point  is  not  altered  hy  any  change  in  ^(.0) 
or  '^{.r)  which  does  not  aflPect  the  character  of  the  points  a.  and  /3.. 
It  depends  therefore  solely  u}>on  the  character  of  the  singularities 
cou)poundcd. 

( 'omplications  arise  only  when  there  is  a  second   pair  of  singu- 
larities o:,,  /3,  such  that 

Clearly  the  residtant  singularity  is  then  dependent  upon  both 
j)airs.  Theii'  etfects  may  he  so  superimposed  as  to  create  an  uglv 
singularitv,  or  they  uiay,  on  the  other  hand,  so  neutralize  each 
other  that  7.  is  a  regular  point.  \'ery  sim])le  examples  of  the 
latter  occurr(Mice  can  he  easily  given.  It  seems  pi'ohable  that 
when  7  is  but  once  a  j)roduct  of  an  7.  by  a  /3,  it  must  alwavs  b(> 
a  singular  ])oiiit,  but  this  has  not  yet  been  j)r()ved.  Its  denion- 
sti'atiou  will  gi-eatly  enhance  the  value  and  a])j)licability  of  /A/c/r^- 
//HZ/v/'.v  tlieoi-ein,  tor  then  it  can  be  stated  in  numerous  cases,  not 
what  the  <ingulai'  points  c  if  /"(.')  may  be,  but  what  they  actually  are. 
A  detailed  -tudv  of  the  luiture  of  the  dei)endence  of  the  singu- 
laritv 7_  upon  7.^  and /:^^  would  probal)ly  be  both  interesting  and 
profitable.  />'o/v7  examines  the  case  in  which  7_  and /5  are  poles  of 
aiiv  <irder-,  j>  ami  7,  and  shows  that  7^  is  then  a  j)ole  of  (U'der 
/'  f  7—  1-  I'  <':ni.  furtliermoi-e,  be  easily  shown  that  whenever 
'/    is  a  pole  ol"  the  fiiv-t  ordei',  7^    is  the  same  kind  of  siiigidar  j)oint 


i)ivi:ij(iKNT  81-;rii:s  and  ('ontimi:!)  FUAr'rioNs.   i  il 

as  /3..  l-^»r  .suj>i)()se  that  we  ])iit  ^,  =  1 ,  wliicli  may  he  done  without 
h).s.s  of  ii'ciicralitv.      The  ])i'inei|)al  part  ot"  (/>(.'•)  at  the  poh'  i..^  i^  then 

;;• —    i  " 

aiul  the  eomj)ositi(>n  of"  this  with  "^(.r)  a;ives  for  the  eoi-respoiuling 
component  ot\/'(.'') 

Hence  the  sinunlarities  7..  and  /3.  differ  bv  a  ninltiiilicative 
constant. 

Only  one  other  general  fact  eoneernino;  the  composition  of  sin- 
gnlarities  ,-eems  to  be  known.  Borcl  j)i-oves  that  if  the  fnnctions 
</)(:',')  and  ■*//-(.'•)  are  one-valued  at  o!_  and  13.  respectively,  f{x)  is 
also  one-valued  at  7  .  Thus  when  two  one-valued  functions  are 
com])ounded,  the  resultant  function  is  also  one-valued.  ]>ut 
this  statement,  as  he  himself  points  out,  must  be  correctly  con- 
strued and  will  not  necessarily  hold  true  when  the  singular  j)oints 
of  the  two  given  functions  are  not  sets  of  isolated  points  but  con- 
dense in  infinite  number  along  curves.  To  construct  an  example 
in  which  /(.'•)  in  not  one-valued,  Borcl  makes  use  of  the  fact, 
now  so  ^ve]l  known,  that  the  decision  whether  the  circle  of  con- 
vergence is  or  is  not  a  natural  boundary  of  a  given  series  depends 
uj)oii  the  arguments  of  its  coefficients.  If,  for  instance,  we  take 
the  series 

1  +e'^'x  +  c'^'-'r  +  ••  •, 

which  has  a  radius  of  convergence  equal  to  1,  by  a  proper  choice 
of  the  arguments  ^  the  circle  of  convergence  can  be  made  a  natu- 
ral boundary.      ]*ut  now 

(♦))  V'  1  -.r  =  c,  +  c,  ,>■  +  'V''+  ••  •' 

in  which  the  coefficients  are  necessarily  real.  C'learlv  the  unit 
circle  will  be  a  natural  boundary  for 


112  TITK  BDSTOX  c()LLO(jrir:\r. 

and  f(ir 

Yet  the  function  /'(■'')  Avhieh  i.s  derived  from  these  two  one- 
valued  functions  hy  Jfdddmard's  ])rocess  i.s  tl)e  two-valued  func- 
tion (G)  which  exists  over  the  entire  j)lane  of  .r. 

I  have  dwelt  at  some  length  upon  ILidaiiuird's  theorem  and  its 
conse(juences  because  of  their  evident  interest  and  importance.  It 
is  worthy  of  note  that  for  analytic  functions  defined  by  power 
series  the  first  great  advance  in  the  determination  of  the  singu- 
laritie-;  over  their  entire  domain  has  been  made  by  methods  that 
are  roughly  parallel  to  tho.se  currently  emj)loyed  in  the  considera- 
tion of  their  convergence.  The  convergence  of  series  is  indeed 
too  difiicult  H  (pu'stion  to  be  settled  by  any  one  rule  or  by  any 
finite  set  of  rules,  but  the  methods  of  comparison  with  series  known 
to  be  convergent  have  been  found  to  be  not  only  most  efficient 
but  also  ade(|uate  for  most  ])ractical  purposes.  In  somewhat 
similar  fashion  IfadanK/i'd'.s  theorem  will  determine  the  singular 
]K)ints  of  numerous  functions  by  linking  them  with  other  series, 
of  which  the  singulai'ities  are  known. 

One  (»f  the  simplest  a])plications  of  this  theorem  is  obtained  l)y 
c()m])ounding  a  given  series 

{~)  "„  +  '',.'•  +  cy  +  •  •  • 

with  itself  once,  twice,  •  •  •,  to  m  times.  .Vll  the  singularities  of 
the  resulting  series 

(S)  a;^  +  a[,r  +  (/>-  +  ..•  (/ ^  1  ,  2,   •  •  ■  ,  >//), 

except  |)()>sihly  .r  =  O  and  .'•  =  oc,  are  included  among  the  points 
obtained  liv  multiplving  /  affixes  of  the  singular  ])oints  of  (7) 
among  themselves  in  all  ])ossil)!e  wavs.  If  the  ///  series  (S)  are 
multiplied  each  bv  a  constant  /-^  and   are  th(;n  added,  a  new  seri(\s 

(!t)  r/(r/j  +  '''(>',).'■+  <■{",).>-+  ■■  • 

i-  otitaiiie(l,  in  which  ''(")  denotes  the  j)olynomial  /y  +  ••■-}-  /.■„,""'■ 


DIA^ERGENT  SERIES  AXD  CONTINUED  FRACTIONS.     113 

This  function  lias  no  singular  points  other  than  those  wliicli  are 

possible  for  the  m  series  from   which   it  was  derived.      A\'hen  r 

different  series 

(t^^  +  a^.v  +  ay-  +  •  •  • , 

6,,  +  6|.K  -4-  b^v^  +  •  ■■  , 

'■„  +  'v«  +  ^v"'  +  ■••, 

are  used^  a  similar  conclusion  is  reached  for  the  series 

^(«o^  ^0'  •  •  •  ^  '"o)  +  ^'(«i^  ^h  •  ■  •?  '"i)-^'  +  G{(U,  b.,,  ■  •  ;  r^)x^  +  ■■■  , 

where  G  denotes  a  polynomial  in  which  the  constant  term  is  lacking. 
These  results  are  of  particular   interest  when   applied   to   the 
series 

(10)  1  +  a;  +  2a;2  -f 1-  nof  +  •  •  • 

and 

(11)  1  +  X+  .,+•••+     -+  •••, 

W'hich  are  the  expansions  of  1  -f  x/^l  —  »;)'  and  log  (1  4-  a').  Since 
these  functions  have  only  one  singular  point,  cc  =  1,  in  the  finite 
plane,  the  only  possible  singularities  of 

are  x=  0,  1,  oo.* 

The  continued  repetition   of  the  above   ])rocess  for  combining 
series  leads  naturally  to  a  consideration  of  series  of  the  form 

(12)  2^("„K 

in  which  a  convergent  power  series  -P(^')  appears  in  place  of  the 
polynomial    G'(m).       Various  theorems  concerning   cases  of  this 

*  Obviously  a  constant  term  can  be  included  now  in  the  polynomial  Gin,  l/«). 


114  THE  BOSTON  coLi/x^nr.^r. 

series  liave  been  oiveu  r(centlv  l)v  Jjon/"  J.e  Ji'oi//r  Jj^sdiuf,'^ 
LinfJr/n/,i  /o/v/  and  FuJxr,^  tliongli  tlie  ])ro()f  of  some  of  these 
theorems  has  no  dii'eet  relation  to  JI(i>hiiiiai''1\s  theorem.  Tlie 
importance  of  such  \vork  is,  hdwever,  a]i])ar('nt,  inasmuch  as  nu- 
merous sories  which  occui"  in  analysis  can  ije  jnit  into  the  form 
under  considerati(jn,  as  for  examj)le  S(sin  tt  n)j''\ 

Three  cases  must  be  distinguished  according  as  the  radius  of 
convergence  of  the  initial  series  (7)  is  less  than,  ecjual  to,  or 
greater  than  1.  If  the  radius  is  less  than  1,  the  singular  point 
nearest  to  the  origin  has  a  mcjdulus  less  than  l,and  the  continued 
multiplication  of  the  afBx  of  the  point  by  itself  gives  a  series  of 
points  Avhich  ap])roach  indefinitely  close  to  the  origin.  The  pre- 
sum])tion,  therefore,  would  naturally  be  that  the  series  (1'2)  is  then 
divergent,  but  this  is  very  far  from  being  always  true,  as  will  be 
seen  at  once  by  rel'erring  to  the  series  2(.r"  siiw/Jand  2(.7'"  cos  a  J 
in  Mhich  <"'„  is  real.  The  a])])lical)ility  of  7/'/'/r///((//(/'.s  theorem 
conse(juently  ceases. 

The  case  in  which  the  radius  of  convergence  of  (7j  is  greater 
than  1  has  been  inve-tigated  very  recently  l)y  Desainf.  In  this  case 
the  expected  theoi'cm  is  obtained.  If,  namely,  /"*(")  is  a  conver- 
gent series  without  a  constant  term,  11^{fij.c"  defines  a  function 
which  can  have  no  singular  ])(;ints,  Ix'sides  .r  =  D  and  .r  =  oc, 
than  those  which  result  from  the  nudtij)lication  of  the  affixes  of 
tlie  singular  j)oint>  among  themselves  in  all  })o>sible  way<  and  to 
any  number  of  times."  *  J)tf;<iiiif's  proof  is  based  upon  the  fact 
that  1/^{iij:\  after  the  omission  of  a  suitable  number  of  terms, 
can  be  exj)ressed  in  the  firm 

*  J'.urn.    'h:  Mnth.,   siT.   •'>,    v,,l.   .",    (]>'.,•'.»),   p.   ."'lo. 
t  I.'ic.  rit. 

i  .['■\iri(.  <h  M'lth.,  M-r.   ."),  vol.  S  (]'.*n2j,  \).  4:'.:;, 

{  Ann   Surirtnli-:    Srltnti'Tui,,     F.hnirn,    vnl.    ol    lllMIl'). 
Jnuin.   ,h  M.,fl,.,   MT.   .'),   vol.   9   iVJiC,,.   [,.   -J-j;]. 

•  M.iih.  Ann.,  vol.  .".:  I  r.Mi;;.,  J).  ;!0'.t. 

^*  I  hi-  ie  :i  -oiiiLwlirit  sharper  stiitcinent  of  the  rtsult  than  that  t;iv(ii  hy  He- 
saint.  In  \i\>  theorem  .<  1  i-  ;.'iven  a^  u  [>o-~ililf  -in^'ular  jn'int,  hut  thi-.  as 
aj'pear';  from  the  pronifto  he  ^iven  here,  is  due  .-olely  to  tlie  admission  of  n  eon- 
.-tant  teriii  into  I'^n  ,.      Hf    also  faiN  to  iKjte  that  r  --  (.1  may  hen  siiiLrular  point. 


i)iVEiK;r:xT  seeiks  and  coxtixikd  fractions.   115 


in  which  _/'(/')  is  the  function  defined  l)y  (7)  for  x  =  t,  c'  is  an  ap- 
projtriately  chosen   contour,  and  c^  denotes  the   ?(th  coefficient  of 

F(u)  =  c^H  -f  ejr  +  ■  ■  •  . 

Although  his  proof  is  essentially  simple  in  character,  I  shall  give 
here  a  new  and  simpler  proof,  based  directly  upon  HadainarcCs 
theorem. 
Place  first 

fj^  =  ai^  +  <r,x  +  u'^^+  ...       (/=2,  3,  ...), 
and  consider  the  expression 

in  wliich  /(  denotes  some  fixed  integer.  If  r  >  1  denotes  the 
radius  of  convergence  of  the  fundamental  series  (7),  the  radius  of 
/,(.r)  will  be  r\  Describe  about  the  origin  a  circle  (?•')  having  a 
radius  r'  <  r".  If  a  sufficient  number  of  initial  terms  be  cut  off  in 
each  of  the  series, 

the  maximum  absolute  values  of  the  remainders  within  or  upon 
the  circle  (/•')  can  be  made  as  small  as  is  desired.  Suppose  then 
that  after  m  terras  of  each  have  been  removed,  the  remainders 

(13)  r,X4      'WiW.      ■■■>     >-M 

do  not  exceed 

respectively,  in  which  e  is  some  small  positive  number.  Let  us 
now  substitute  in  Hadamard''s  integral 


116  THE  BOSTON  COLLOQUIUM 

1 


any  two  of  tlie  functions  (13)  for  (f)  and  ^|r. 
Put  for  example 

1\  /I 


<I^M='\.-rM,  ^(.^) 


and  choose  the  unit  circle  as  the  path  of  integration.  Then  if 
1 2;  ]=/•',  the  absolute  values  of  the  arguments  of  the  series  (/>  (233) 
and  -v/r^l  /.c)  will  be  less  than  their  radii  of  convergence  since 
\x\=  1  and  /•>  1.  The  conditions  for  the  existence  of  IFada- 
mar(l''s  integral  are  therefore  fulfilled.     Since  also 


■\-.(^^ 


we  have 


-(^)i-^/'m'-'- 


But  by  JIaddinard's  theorem  F{z)  =  )\^^_^._^.{z),  and  hence 

(1-^)  l'-.(^)l<^'  (l^l^O, 

for  all  values  of  /  from  2ii  to  An  inclusive.  The  reasoning  can 
now  be  repeated  with  2//  in  j)lace  of  71,  and  so  on  ;  therefore  (14) 
is  true  for  all  values  of  i^  n. 

Thus  far  the  value  of  e  has  remained  arbitrary.  Let  its  value 
now  be  taken  less  than  the  radius  of  convergence  of  i^(")-  Then 
by  ( 1  4)  the  series 

(1^)  '•/„(•'•) +  'VH'Vf,(-^)  +  --- 

will  be  uniformly  convergent  in  (/•').  Since,  furthermore,  all  the 
comjxdiciit  sei'ics  '■„.,('  =  '*,  1,  -,  •  •  •)  are  likewise  so  convergent, 
by  a  fundamental  and  familiar  theorem  of  ]\'<'ifr.sfr<iss  *  the  terms 
of  the  collective  series  (lo)  may  be  rearranged  into  an  ordinary 
seri<'s  in  ascending  jiowers  of  x.      J>ut  this  rearrangement  gives 

'  1  l:irkiic--s  ami  MnrK'v's  Inlrn'liirlion  In  thr  'T hi-onj  nf  AnolijUr  Functions,  p.  134. 


DIVEllGEXT  SERIES  AND  COXTIXrED  FRACTIONS.     117 

Z  (  Z  <■,"■  )''', 
or  tlie  reniaindt'r  after  the  Qn  —  l)th  power  of  x  in 

7  =  0 

Now  the  series  (15^  before  its  rearrangcraent  was  a  uniformly 
convergent  series  of  analytic  functions  and  defined  a  function 
which  was  analytic  within  (/•').  It  follows  that  (16)  is  also 
analytic  within  this  circle,  and  hence 

has  no  singularities  within  this  circle  except  those  of 

But  the  radius  of  (/•')  was  any  quantity  short  of  /•",  and  this  con- 
clusion therefore  holds  within  a  circle  having  its  center  in  the 
origin  and  a  radius  equal  to  /■",  By  increasing  n  indefinitely,  the 
theorem  of  JJcsainf  results.  It  is  evident  also  that  if /^(.r),  and 
therefore  ,/';(.'.'),  represents  a  one-valued  function,  '^P(ayii"  must 
also  be  such  a  function. 

There  remains  yet  for  consideration  the  third  class  of  cases  in 
wiiich  the  radius  of  convergence  of  the  fundamental  series  is  1. 
If  upon  the  circle  of  convergence  there  is  any  singular  point  with 
an  inconinicnsurable  ai'giunent,  the  continued  midtiplication  of  its 
affix  by  itself  gives  a  set  of  ])oints  everywhere  dense  upon  the 
circle  of  convergencse.  It  is  therefore  to  be  expected  that  this 
cin^le  will  be,  in  general,  a  natural  boundary  for  S7''(a J.>f,  and 
accoi'diugly  the  cases  which  will  l)e  of  chief  interest  will  be  those 
in  which  all  the  singular  points  u[)()n  the  circle  have  commensur- 
able arguments.  A  simple  case  of  this  character  is  obtained 
when  either  {\())  or  (11)  is  chosen  as  the  generating  series.  If 
the  former  be  selected,  the  resulting  series  has  the  form  w7^(»)x". 
This   has  a  special   interest   inasnuich  as  its  study  has    proved   to 


118  THE  BOSTON  COLLOQUIUM. 

be  of  profit  both  for  the  theory  of  analytic  continuation  and  of 
divergent  series.  Tlie  reason  becomes  apparent  when  the  state- 
ment is  made  that  it  is  })ossible  to  throw  any  Taylor's  series, 
'S.dx",  whether  convergent  or  divergent,  into  the  particular  form 
SP(?i)ft;",  and  in  an  infinite  number  of  ways.  This  fact  follows 
as  a  corollary  of  a  very  general  theorem  of  Mittag-Leffler,*  which, 
when  restricted  to  the  special  case  before  ns,  establishes  the  exist- 
ence of  a  function  P{x),  which  is  holomorphic  over  the  entire 
finite  plane  and  assumes  the  pre-assigned  values  a^,  (t^,  a.„  ■  ■  •  in 
the  points  o:  =  0,  1,  2,  •  •  •.  Consequently  the  character  of  the 
function  defined  by  2/-*()))a)"  is  made  to  depend  upon  the  behavior 
of  I^{x)  as  03  aj)proaches   oo. 

Inasmuch  as  Si^(n)a;"  is  perfectly  general,  limitations  must  be 
imposed  upon  P(u)  in  any  attempt  to  extend  IIa<l<niiar<Vs  theorem 
to  this  series.  But  whenever  the  theorem  is  applicable,  the  only 
possible  singularities  of  I.I^nyx"  are  x  =  0,  1,  oc.  Lecni  f  estab- 
lishes the  correctness  of  this  result  when  J^{u)  is  an  entire  function 
of  order  less  than  l,^;  giving  also  a  more  general  theorem  >;  concern- 
ing Si-*(a,Jx"  of  which  this  is  a  special  case.  The  like  conclusion 
holds  concerning  the  singular  points  of  S^(l/?i)x",  provided  only 
that  I-*(x)  is  holomorphic  at  the  origin. || 

Very  recently  these  results  of  Lecui  have  been  proved  more 
simply  by  Faher,  but  in  a  more  restricted  form,  an  artificial  cut 
being  drawn  from  :k  =  1  to  .''  =  oo  to  obtain  a  one  valued  func- 
tion. In  addition,  Faher  shows  that  if  for  any  prescribed  e  and 
for  a  sufficiently  large  /•  the  ine(juality 

(17)  \P{rc''i>)\<e''- 

*  Arid  Mh/Il,  vol.  1  (lSS-1),  ]).  0.'),  thccirem  I).  Fora  reference  totliis  tlieoi'eni 
I  am  indebted  to  Professor  ( )sjj;o()d.  Theorem  2  of  Dcsaint's  memoir  ( ]i.  4.'58) 
is  in  contradiction  with  this,  l)ut  his  ])roof  is  liere  inadequate  since  i\  (p.  4  10)  lias 
not  necessarily  a  lower  limit. 

\  Lor.  ril.,  !>.  -lis. 

X  He  alsci  shows  that  -/'(/*)'"  is  then  a  one-valued  t'unt'tion. 

U>"''.  r/C,  p.  117.  See  also  /;/;//.  <!,•  In  Son.  Malh.  <h  FniiLcc,  vol.  -Jti  (IS'.)S), 
],.  2t;7. 

"  L',i\  rit.,  p.  .lis  ;  see  alx)  p.    Id?. 


i)ivi:k(;ext  seuies  and  t'OXTixrFr)  fuactioxs.   il!> 

is  rultilled,  the  j)oiut  x  =  L  must  be  an  essential  singularity,  and 
the  function  represented  by  '^P(^n)x"  is  C()nse(|U(Mitly  one- valued.* 
(■onversely,  if  t\x)  is  a  one-valued  function  which  has  only  one 
singidar  point,  and  if  that  point  is  an  essential  singularity, _/'(.c) 
can  be  expressed  in  the  form  SP(;;)x'",  in  which  I^(u)  is  an  entire 
function  satisfying  (17).  More  generally,  if  there  are  /essential 
singularities  x^,  ■  ■  •,  x^  and  no  other  singular  points  in  the  finite 
plane,  the  coefficient  of  .c"  must  be 

•'  1  ; 

in  which  P^i't),  ■  •  -,  -^,('0  ^''^  entire  fiuictions  of  the  nature  above 
specified  and  in  which  lini  ;  «_]  =  0.  This  converse  has  an  espe- 
cial interest  because  as  yet  few  theorems  have  been  discovered 
giving  the  necessary  form  of  the  coefficients  of  a  power  series  for 
an  analytic  function  with  prescribed  functional  properties. 

Other  theorems  concerning  'EP(n).v"  have  recently  been  derived 
without  requiring  that  P{it)  shall  be  holomorphic  over  the  entire 
plane. 

As  a  sample  of  these  I  shall  cite  in  conclusion  the  following 
theorem  of  LindeVof:  t 

If  P{ii)  represents  a  function  fulfilling  the  following  conditions  : 

1.  Piji)  is  analytic  for  every  point  of  the  complex  plane 
z=^T-\-it  for  which  t^O  (except  possibly  at  the  origin,  for 
which  P{z)  has  a  determinate  value). 

2.  A  number  c  being  arbitrarily  given,  it  is  possible  to  find 
another  numl)er  7^  such  that  by  putting  ;,  =  re"'-'  we  will  have  for 
r  >  R 


*  Le  Roy  tliree  years  earlier  liad  noted  this  eonelusioii  wlien  /'(.')  is  an  entire 
fiineiion  wliose  "apparent  order"  is  less  than  1  ;  Inc.  clt.,  p.  o-ls^  footnote. 
Faher  does  not  seem  to  he  aware  of  I>e  Hoy's  statement.  The  ditierenee  Ix^twcen 
the  two  statements  is  slight  hut  heeomes  important  in  fornndating  the  new  and 
interesting  converse  which  Faher  add.«. 

i  hoc.  r!t.,  i  i:;. 


ll^<»  THE  BOSTON  COLLOQUIUM. 

then  the  principal  i)ranch  of  the  fnnetion  S7^(7()ri"  Avill  be  liolo- 
morphic  tliroughont  the  coni])lex  j)hinc  excepting  poss^ibly  on  the 
segment  (1,  +  'X>  )  of  the  real  axis.  Furthermore,  the  function 
ap})roache.s  0  a.s  a  limit  when  .r  tends  toward  the  point  at  infinity 
along  any  rav  having  an  arirument  between  0  and  27r. 

Lectl'RK  IV.      On   Scries  of  Pohinoiaiak  and  of  Rational 

Fractions. 

In  the  last  two  lectures  I  have  spoken  of  the  use  of  integrals  for 
the  study  of  analytic  extension  and  of  divergent  series.  The  topic 
of  to-day's  lecture  is  the  representation  of  functions  by  means  of 
series  of  polynomials  and  of  rational  fractions.  This  subject  forms 
a  very  natural  transition  to  the  succeeding  lecture  upon  continued 
fractions,  since  an  algebraic  continued  fraction  is  in  reality  noth- 
ing but  a  series  of  rational  fractions  advantageously  chosen  for  the 
study  of  a  corresponding  function  which,  when  known,  is  com- 
monly given  in  the  form  of  a  power  series. 

The  literature  relating  directly  or  indirectly  to  series  of  poly- 
nomials and  of  rational  fractions  is  a  vast  one,  with  many  ramifi- 
cations. Thus  in  one  direction  there  are  various  researches  of 
imj)ortance  upon  the  non-uniform  convergence  of  series  of  contin- 
uous functions,  and  in  this  connection  I  mav  refer  particularly  to 
the  recent  work  of  ( hr/ood  and  Baire,  an  excellent  rej)ort  of  which 
is  contained  in  SdionflivH'  Ihriclit  I'lher  die  Jfcjn/cnlchr''.*  An- 
other part  of  the  lield  comprisi'S  numerous  memoirs  devote<l  to 
special  series  of  polvnomials  and  rational  fractions,  (^uite  re- 
cently a  more  systematic  and  general  studv  has  been  begun  l)y 
/,'o/v/,  Mi/fn>/-/,rjilrr,  and  others,  and  it  is  to  tliis  that  J  am  to 
call  voin'  especial  attention. 

Two  veiv  familiar  facts,  both  discovered  by  HV/r/'.sY/Y/.s.s',  may 
be  -aid  to  l>c  the  origin  of  this  studv.  I  refer,  of  cours(>,  to  the 
tlicorcrn  that  anv  t'unctioii  which  is  continuous  in  a  gi\'en  linite 
intei'vai  of  the  I'eal  axis  can    l)e  expressed  in    that    interval    as   an 

■  .Jahr(.<l,.-rirht  .1,1-  d,,il.«-h.„  M.,th.  ,„nt,hn -Vrrriinuinri,   vul.   S,   ,,,,.   •_'-Jl--Jll. 


DIVEKGENT  SERIES  AND  CONTIXEED  FRACTIONS,     i -' 1 

absolutely  and  uniformly  couvcrt^ent  scries  of  ])()lyuoiuials/''  and, 
secondly,  to  the  possibility  that  a  single  series  of  rational  fractions 
may  I'cpresent  two  or  more  distinct  analytic  functions  in  different 
portions  of  its  domain  of  convergence.  A  notable  advance  nj)on 
the  theorem  first  mentioned  was  made  by  Run(/e\  in  1884,  who 
proved  that  any  one- valued  analytic  function  throughout  the  do- 
main of  its  existence  can  be  represented  by  a  series  of  rational 
functions;  furthermore,  this  domain  may  be  of  any  shape  what- 
soever, provided  only  it  forms  a  two-dimensional  continuum. 
Run<ji:\  jiroof  of  these  important  results  is  not  only  worthy  of 
careful  study,  but  contains  also  certain  conclusions  which  were  an- 
nounced again  by  PainleveX  in  1898,  though  without  proof. 
The  conclusions  reached  were  as  follows  : 

Let  I)  be  a  domain  consisting  of  any  number  of  separate  pieces 
t)f  the  complex  plane,  in  each  of  which  we  will  suppose  an  analytic 
function  to  be  defined.  The  functions  thus  defined  can  be,  at 
})leasure,  either  distinct  functions  or  parts  of  one  or  more  func- 
tions. In  any  case  a  series  of  rational  functions  can  be  formed 
which  will  converge  absolutely  and  uniformly  in  any  region 
l\ing  in  the  interior  of  I)  and  rej)resent  in  each  separate  piece 
the  prescribed  function.  Furthermore,  this  representation  can 
be  made  in  an  infinite  number  of  ways.  I^et  the  ensemble  of 
the  points  excluded  from  J)  be  represented  by  K.  When  A' con- 
sists of  a  single  connected  continuum  of  any  sort,  whether  linear 
or  areal,  any  point  a  of  K  can  be  arbitrarily  selected,  and  the 
function  can  be  expanded  into  the  series 

^    "  \x  —  a 


"•■•  "  I't'bcT  (lie  analytische  Darstelll)iirkeit  sogenannter  willkiirliclier  Functionen 
fint-r  leellcii  X'eraiulerliclien '' ;  Bi'rliniT  SilziiiHj.-ihirirhte,  ISS-l,  p.  ():>,']  or  W'erLr, 
vol.  I),  p.  1.  Sini})l(j  proofs  of  tlie  theorem  have  heeii  given  Ijy  Lebesque,  Jhill. 
,!,'.•<  .Sr/V/ir,,.-  .!/((//,.,  ser.  2,  vol.  22  (IHU^),  )>.  278,  and  by  Mittag-Lefller,  IhiKUnrnti 
(li  PdUrnui,  vol.  14  (I'.MJO),  p.  217,  with  an  extension  to  functions  of  two  variables. 
Jn  this  connection  see  Painleve's  note  in  the  Cvuipt.  licmL,  vol.  ]2(')  (1898),  p.  Ah^X 

i  Aftn  Mnih.,  vol.  t;,  p.  229. 

tOnnpt.  JU',1'1.,  vol.  12G,  ])p.  201  and  318. 


1^2  T]IK  BOSTOX  COLLOQUIUM. 

in  which  T/^,  [1  /(.''  —  ")']  (leiiotes  a  polynomial  in  1  /(x  —  a).  If,  in 
particular,  the  continuum  I'J  contains  tlic  point  a^  =  oc,  an  ordinary 
scries  of  polynomials,  '2(rjx),  can  be  employed.  AVhen  ^con- 
sists of  a  finite  number  of  separate  pieces  (or  isolated  j)oints),  the 
expansion  can  be  put  under  the  form 

t^'H  ^  A  +  t(^T(    ^  -)  +  ■■■+ f:<^:r(    ^ 

in  which  a^  •■•,(',  are  ])oints  arl)itrarily  chosen  in  the  separate 
pieces. 

In  the  familiar  case  in  which  only  a  single  analytic  function 

(1)  «^  +  «j(.f  —  a)+  (t,{x  —  a)-  +  .  •  . 

is  given,  it  is  natural  to  seek  a  series  of  polynomials  having  the 
greatest  possible  domain  of  convergence.  Unless  the  function  is 
one- valued,  the  most  convenient  domain  is  in  general  the  star  of 
3Iift(i(/-LeJf{er.  This  is  constructed  for  the  series  (1 )  by  first  marking 
on  each  ray  which  terminates  in  a  the  nearest  singular  point  and  then 
obliterating  the  portion  of  the  ray  beyond  this  point.  The  region 
which  remains  when  this  has  been  done  is  a  star  having:  ((  for  its 
center.  ^fiff((f/-Lelfl<•r*  shows  that  within  the  star  the  given  ana- 
lytic function  can  be  represented  by  a  series  of  polynomials  in 
which  the  coefficients  of  the  ])olvnomials  de]>end  onlv  upon  the 
value  of  the  function  and  its  derivatives  at  a,  t  or,  in  other  words, 
uj)on  the  coeflicients  of  (1).      [f,  in  short,  we  put 


.''/»-ZZ---E 


>,    +    \+---    +    \)-"n 


X, :  X, : ■  ■  \  !       V    " 


X  —  (t 


■  An,,  Math.,  v,,l.  ■2:\  ( isitui,  p.  i:; ;  vd.  I'l.  pp.  is:;,  'ji),-);  vol.  -J';,  j..  :;'i:;.  A 
t,'o()i|  --uniinary  is  t'uuml  in  tlic  I'mr.  ,,/  ihr  Londaii  M,it/i.  .S'oc,  vul.  :>'2  (  I'.tndi,  p]). 
7  H  ^  7  *^ 

i  In  tills  i-i-s]irct  his  work  is  siijirrior  to  that  of  riniit,'*'  and  otliors.  Itmi^'e, 
for  example,  |ire~ii]ip  isf>  a    knowlcijijf  of  the  fimctioii   at  an    iiilinile   niini'ier  of 

lioi  Ills. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     123 

then  ^  ^^.O^O  i'^  ^  ^series  which  converges  uniformly  in  any  region 

lying,  with  its  boundary,  entirely  in  the  interior  of  the  star.  The 
series  may  also  converge  outside  the  star.  Borel  *  has  shown, 
furthermore,  that  the  series  of  Mitta(j-Le§icr  is  not  the  only  possi- 
ble one,  but  there  is  an  infinity  of  polynomial  series  sharing  the 
same  property  within  the  star. 

It  will  be  noticed  that  the  construction  of  the  series  of  Miitag- 
Leffier  is  iu  no  wise  dependent  upon  the  convergence  of  the  initial 
power  series.  In  certain  cases,  at  least,  the  polynomial  series  con- 
verges when  the  given  series  (1)  is  itself  divergent.  It  is  natural 
tlierefore  to  look  for  a  theory  of  divergent  series  based  upon  con- 
vergent series  of  polynomials.  As  yet,  however,  no  such  theory 
has  been  invented.  One  of  the  chief  difficulties  in  the  way  is  that 
the  ])olynomial  series  do  not  afford  a  unique  mode  of  representing 
an  analytic  function.  Xow  the  difference  between  any  two  series 
of  polynomials  for  the  same  function  in  an  assigned  area  is  a  third 
series  which  vanishes  at  every  point  of  the  area,  though  the  sej)- 
arate  terms  do  not.  This  is  a  decidedly  awkward  point,  and 
occasions  difficulty  in  proving  or  disproving  the  identity  of  two 
functions  expressed  by  polynomial  series.  It  is  true,  indeed,  that 
this  difficulty  will  scarcely  present  itself  when  we  start  with  a  con- 
vergent })ower  series  which  is  to  be  continued  analytically,  the 
polynomial  series  then  giving  continuations  of  a  common  function. 
But  when  the  series  (1)  is  divergent  and  there  is  no  known  func- 
tion which  it  represents,  it  is  an  open  question  whether  the  differ- 
ent series  of  polynomials  which  are  obtained  from  (1)  by  a])plica- 
tion  of  diverse  laws  will  furnish  the  same  or  different  functions. 
If  different  functions,  is  there  any  ground  for  preferring  one  series 
of  polynomials  to  another? 

Up  to  the  pres(Mit  time  two  essentially  different  principles  seem 
to  have  been  followed  in  the  formation  of  series  of  polynomials. 
In  the  work  of  Ji'mifj",  J>ore/,  Pdinlcrc  and  Jfi(f(i(/-L(iih'r  the  co- 
efficients  in  the  polynomials  vary  with   the  character  of  the  ana- 

^Ann.  lie  I'  !■:<■.  Xo,-.,  ser.  2,  vol.  ]C>  (KSSni).  p.  ]:V2,  or  L>'.<  Svrif.^  illirnieutes, 
p.  171. 


124  THK  BOSTON  COLLOQUIUM. 

lytic  function  to  he  repro:^cntc(l  ;  for  example,  in  the  polynomials 
of  Mlff(i;/-Lf'lff('r  they  are  functions  of  the  coefficients  of  the  given 
element,  ^'''„-'"".  By  ap])ropriately  choosing  the  coefficients  of  the 
j)olynomials  these  writers  obtain  a  very  large  region  of  conver- 
gence anil  at  the  same  time  are  able  to  greatly  vary  its  shaj)e.  On 
the  other  hand,  the  series  which  are  met  in  the  practical  branches 
of  mathematics — for  instance,  in  the  theory  of  zonal  harmonics — 
have  the  f  )rm 

in  which  the  polynomials  G^^(x)  are  entirely  independent  of  the 
fnnction  represented,  while  the  e.  vary.  The  polynomials  them- 
selves are  selected  according  to  the  shape  of  the  region  of  con- 
vergence.     Thus  if  the  region  is  a  circle,  we  may  put 

r;,(.'t')  =  (.3  -  a)", 

and  we  liave  then  the  ordinary  Taylor's  series.  Or  if  it  be  an 
elli]>se  having  the  foci  ±  1,  we  may  take  for  our  polynomials 
either  the  successive  zonal  harmonics  or  a  second  succession  of 
polynomials  (also  called  Jjecjendre^s  polynomials)  which  are  con- 
nected by  the  recurrent  relation  : 

{■.\)        </„.,(.-)  -  2,'{;ln  +  ::!)(7_,(:'-)  +  4(r  -f  1)^0  (^0  =  0. 

In  a  recent  number  of  the  M'<iflir)ii(if!.-;clir  AiukiIcii  (July,  1903) 
I'ahrr  has  considered  this  second  class  of  polynomials  from  a  some- 
what general  point  of  view  and  has  demonstrated  that  any  function 
whieli  i-  holomorj)hic  within  a  closed  in'anch  of  a  single  analytic 
<'ur\e,  a-  f )r  ex;un]»le  an  ellipse  or  a  leinniscate  of  cnie  oval, 
can  be  expressed  by  a  series  of  the  foi'm  (2).  The  pro])erties 
of  hi-  -erie.s  are  similar  to  tho^e  of  Tavlor's  s(M'ies.  In  the 
ca-e  ol'  the  latter,  to  aseei'tain  whether  '^c  y  converges  in  the 
iiitei-idi'  of  a  cii'eje  having  its  center  in  the  origin  and  a 
ra(liu~  /i',  we  have  oiilv  to  det<'i'niine  the  iiiaxiinum  modulus  of 
a  point  of  CO  11  den --at  ion  of  tlx?  set  of  points  i  a^^  {  n  =  1 ,  2,  ^),  •  •  ■), 
if  it  i-  exaetiv  e(Mial    to    1    /,',  the   eirele  (  /.')  is    the  circle  of  con- 


DIVIOKGENT  SKHIES  AND  ("OXTINUKD  FKACTIOXS.     125 

vergeiice,  and  there  h  at  least  one  singularity  u))()n  its  circinnfer- 
ence.  If,  on  the  other  hand,  it  is  greater  or  less  than  1  li,  the 
series  will  have  a  smaller  or  a  larger  circle  of  convergence.  So  also 
to  the  given  brancli  of  the  analytic  curve  there  corresponds  a 
certain  critical  value.  When  this  is  exactly  equal  to  the  upper 
limit  of  j  1  cj  in  Faher^s  series,  the  given  analytic  branch  is  the 
curve  of  convergence.  At  every  point  within,  the  series  converges, 
while  it  diverges  at  every  exterior  point,  and  upon  the  curve  there 
must  lie  at  least  one  singular  ])oint  of  the  function  defined  by  (2). 
If,  however,  the  upper  limit  is  greater  or  less  than  the  critical 
value,  we  consider  a  certain  series  of  simple,  closed  analytic  curves, 
(as  for  example  a  series  of  confocal  ellipses),  among  which  the  given 
analytic  branch  must,  of  course,  be  included.  The  curve  of  con- 
vergence is  then  fixed  by  the  reciprocal  of  the  upper  limit  of 
I  ]  (' J  provided  this  limit  is  not  too  large.  ^Moreover,  as  in  the 
case  of  Taylor's  series,  the  function  cannot  vanish  identically  un- 
less every  c,^  =  0,  and  in  consequence  the  series  vanishes  identi- 
cally. It  is  therefore  impossible  that  the  same  function  shall  be 
represented  by  two  different  series  of  the  given  form. 

In  view  of  the  last  mentioned  fact  it  might  be  of  es])ecial  inter- 
est to  apply  this  class  of  polynomial  series  to  the  study  of  diver- 
gent series. 

In  the  most  familiar  and  useful  polynomial  series  the  successive 
polynomials  are  connected  by  a  linear  law  of  recurrence, 

(4)         ^•o^,...(-'-)  +  K  G.„U-r)  +■■■  +  ^'-^^.C^O  =  0, 

in  which  the  coeflficients  l\  are  polynomials  in  x  and  n.  Thus  the 
zonal  harmonics  have  as  their  law  of  recurrence 

Many  series  of  this  nature  are  also  included  in  the  class  con- 
sidered by  Faher.  The  form  of  the  region  of  convergence  has 
been  determined  by  Poineare  *  upon  the  hypothesis  that  equation 

*  Amer.  Journ.  of  Math.,xo\.  7  (1885),  p.  243. 


126  THE  BOSTON  COLLOQUIIM. 

(4)  has  a  limiting-  form  for  ?;  =  oc.  I^ct  the  equation  be  first 
divided  througli  l)y  L\^,  and  then  denote  the  limits  of  the  successive 
coefficients  for  n  =  oc  by  l'\{->'),  /'■.,(^),  ■  •  /•'„,(•'')•  ('onstruct  next 
the  auxiliary  equation 

(5)  .'"  +  I:,(x)z""'  +  k^')z"'-'  4-  •  •  ■  +  /•i.r)  =  0. 

Except  for  particular  values  of  .r  there  will  be  one  root  of  this 
equation  which  lias  a  larger  modulus  than  any  other.  I^et  r(x) 
be  that  root.  PoinGare*  shows  that  with  increasing  n  the  ratio 
G, (.(')/ fi'„_,(.r)  will  approach,  in  general,  this  root  as  its  limit. 
The  region  of  convergence  is  therefore  confined  by  a  curve  of  the 
form  C=  i/'(^)|,  and  the  value  of  C  for  the  series  (2)  is  to  be 
taken  equal  to  the  radius  of  convergence  of  2c^7/".f 

By  way  of  illustration  let  us  take  the  series  2c  (7  (x)  in  which 
the  polynomial  obeys  the  law 

*More  sj)ecifica]ly,  Poincare  proves  tliat  if  no  two  roots  of  (5)  are  of  equal 
niodiilus,  6'„(.r):'G„_i(r)  lias  always  a  limit,  and  this  limit  is  equal  to  some  root  of 
(5),  usually  tlie  one  of  greatest  modulus. 

t  Poincare  has  given  no  proof  that  the  series  (2)  will  converge  at  those  points 
within  the  curve  |  r{r)  \  =  C,  for  which  there  are  two  or  more  distinct  roots  of 
(5)  having  a  common  modulus  greater  than  the  moduli  of  the  remaining  roots. 
Thus  in  the  example  which  is  quoted  below  (p.  127),  these  are  the  ])oints  of  the 
real  axis  which  are  included  between  -\  1  and  —  1.  Tliis  gaji  in  Poincare's  theory 
can  be  filled  in  by  the  following  theorem  which  1  have  given  in  the  Transaction-H 
of  the  Amcr.  Math.  Soc,  vol.  1  (litOO),  p.  '20S  :  If  the  coeflicients  in  the  series 
i^.l,,//"  are  connected  by  a  recurrent  relation  having  the  limiting  form 

.4„  +  /,-,.l„_,  4- •■•+/.■„, .T,-,„=0, 

tlie  scries  will  converge  at  tlie  worst  witliin  a  circle  wiiosc  radius  is  the  recipro- 
cal of  tlic  greatest  modulus  of  any  root  of  the  auxiliary  e(juation 

z"  +  k,z'-'] i/„,-0. 

I)('n()tc  thi--  niaxiinuni   by  r,  inr.<j>rptiir  at'  the  nvmlier  af  r<iiit.<  Imrinij  thi<  inaxhunn 

I.T,  i<.V(r   ;    0"  ("-1,  2,  ...  ). 

Hence  if  <'\^  tlic  radium  of  convergence  of  -e„ii'\  tlic  scries  '^e„An  will  converge 
when  >'  ■/.  Sup|lO■^e  iio\\  lli:it  >■!.-  (U']iend^  ii]i()ii  ./  and  put  A,,  ^^-  (i,(j)-  It 
t'i.llnw>  then  Iroin  my  tlienrcm  tlnit  1r,,( , „[y)  will  n/i/wi/,^  ronverLie  when  C^  r. 
r.nt  till-  i-  \\li:il  \vM-  to  be  I'mved. 

.\t  the  time  lit' (lie  publicalidii  of  my  work  I  w;is  not  aware  of  I'oinc;ire'.->  article, 
:in(l  1  tlieret'ire  failed  to  ]>oint  out  tile  relation  of  the  two  memoirs. 


DTVEKGEXT  SEKIKS  AXI)  CONTIXIKI)  FlfACTIOXS.     127 

(,r  +  i)6;,../.'-)  -  2<nr(;^^. ,(..•)  +  ("-  +  -'-jOp')  =  0. 

For  ,/  =  X  the  limiting'  form  of  this  equation  is 

or  thie  same  as  the  limiting  form  for  the  zonal  harmonic.  The 
auxiliary  e(|uation  is 

^  -2xz-i  1  =  0, 

of  which  the  roots  are 

z  =  X  ±:  Vor  —  1. 

The  curves  '.  x  d=  ]  x-  —  1  =  C  are  easily  seen  to  be  ellipses 
liaving  the  foci  ±  1.  Hence  if  JR  is  the  radius  of  convergence 
of  '^cjf",  the  region  of  convergence  of  (2)  is  the  interior  of  an 
ellipse, 

1  X  ±  v'-v^^l  I  =  B. 

Poincare  also  examines  such  exceptional  cases  as  that  which  is 
specified  by  relation  (3),  which  has  no  proper  limiting  form.  But 
upon  this  work  Ave  can  not  longer  dwell.  I  wish,  however,  to 
emphasize  its  fundamental  character,  inasmuch  as  many  previous, 
and  even  sul^jsequent  conclusions  concerning  the  convergence  of 
series  of  the  form  (2)  are  comprised  in  Poinccre's  result. 

Somewhat  earlier  in  the  lecture  I  set  forth  the  arbitrary  charac- 
ter of  the  function  which  could  be  represented  by  series  of  poly- 
nomials and  rational  fractions.  We  have  seen  also  how  this  arbi- 
trary element  was  entirely  eradicated  by  confining  ourselves  to 
polynomials  which  obey  a  linear  law  of  recurrence.  In  the  remain- 
der of  this  lecture  I  wish  to  develop  the  consequences  of  restrict- 
ing a  series  of  rational  fractions  in  the  manner  supposed  by  Bore!  in 
his  thesis  *  and  its  recent  continuation  in  the  Acta  Mcf/ieuKdica.f 
Borel  seeks  to  so  restrict  a  series  of  rational  fractions,  2i^  (.r)//i'^^(.r), 
as  to  ensure  a  connection  between  the  position  of  the  poles  of  its  sep- 
arate terms  and  the  })osition  of  the  singular  points  of  the  function 
which  the  series  collectively  represents.     On  this  account  he  assigns 

*  Ann.  ./»'  /'/>.  So,:,  ser.  3,  vol.  12  (ISit-l),  p.  1. 
t  Vol.  24  (11)00),  p.  309. 


128  THE  BOSTOX  COLLOQX'II'M. 

an  upper  limit  to  the  degrees  of  I\^  (,r)  and  J>J-r).  But  tliis  is  not 
enouii;h,  and  he  proeeeds  therefore  to  limit  the  magnitude  of  tiie  co- 
effieients  in  the  numerators.  On  the  other  hand,  he  allows  any  dis- 
trii)ution  whatsoever  for  the  roots  of  the  denominators,  thus  leaving 
himself  at  liberty  to  vary  greatly  the  nature  of  the  funetion  rep- 
resented. 

In  his  thesis  he  develops  the  case 

which  had  been  previously  considered  l)y  Pobwdre  *  and  Goursat.'\ 
To  avoid  semi-convergent  series  or,  in  other  words,  functions,  of 
which  the  character  depends  not  merely  upon  the  position  of  the 
poles  (i^^  and  the  values  of  ^1^^  ])ut  also  upon  the  order  of  summation, 
the  condition  is  imposed  that  S^l^^  shall  be  absolutely  convergent. 
Then  if  there  is  any  area  of  the  s  plane  which  contains  no  poles, 
the  series  (G)  must  converge  within  this  region.  Since  further- 
more it  is  uniformly  convergent  in  any  interior  sui)-region,  it 
defines  an  analytic  function  within  the  area.  There  may  be 
several  such  areas  separated  by  lines  or  regions  in  which  the  poles 
are  everywhere  dense.  This  is  precisely  the  case  to  be  consid-ered 
now. 

To  simplify  matters,  let  ns  suppose  that  the  poles  are  every- 
where dense  along  certain  closed  curves  of  ordinary  character, 
but  nowhere  inside  the  curves.  Po'iavare  and  Gonr^'ai  show  that 
each  curve  is  a  natural  boundary  for  the  analytic  function  ^(z) 
defined  by  ((])  in  its  interior.  I>orel\s  proof  is  as  follows.  De- 
note the  component  of  (<j)  which  corresponds  to  a^^  by 

and  the  remaining  part  by 

*  Arlit  S.,riitnti.<  Fninirii,   vol.  VI  (ls8:^),  p.   .'?41,  and  Amer.  Journ.  of  Math, 

vol.  11  (  \S'r2i,  II.  2(1 1. 

I  Cninpi.  HikI.,  vol.  Ill  (1SM2),  |i.  715. 


DIVERGENT  SERIES  AND  CONTIXrEI)  FRACTIONS.     12!) 

It  is  evident  that  if  c.^^  lies  within  any  (Mie  of  the  carves  considered, 
(/  is  a  }iok'  of  0(^),  Now  when  these  interior  poles  condense  in  in- 
finite nnniber  in  the  vicinity  of  any  point  of  the  curve,  it  must, 
of  course,  be  a  singularity  of  4>{:c).  Consider  next  any  one  of  the 
points  «^  which  lies  upon  the  boundary  but  is  not  a  point  of  con- 
densation of  the  interior  poles,  and  let  ?:  approach  this  point  along 
the  normal.  Describe  a  circle  upon  the  line  s  —  a  ^  as  diameter. 
If -i  is  sufficiently  near  to  a^^,  the  circle  will  exclude  every  one  of 
the  points  a.,  excepting  a^^  whicii  lies  upon  its  boundary.  Since 
also  S.l^  is  absolutely  convergent,  by  increasing  /■  the  second 
component  of  ^.,('-:)  may  be  made  less  in  absolute  value  than 
ej\z  —  rf  I"',  in  which  e  is  an  arbitrarily  small  prescribed  quantity. 
If,  then,  II  denotes  the  maximum  of  the  first  component  of  (^Jy^-) 
as  z  now  moves  up  to  a^^,  we  have 

Consequently, 

lim  (/)(,^)  •  iji  —  (ly^  =  lini  (/)j(-.)  •  (,^  -  «J"'  +  lim  (/>,(-)  •  {re  —  aJ"'=B^^^. 

This  shows  that  \(f>{^)\  increases  indefinitely  when?.  a})proaches 
any  pole  a  _  of  the  inth  order  along  a  normal,  and  removes  the  pos- 
sibility that  the  poles,  because  they  are  infinitely  thick  upon  the 
curve,  may  so  neutralize  one  another  that  the  function  can  be  car- 
ried analytically  across  the  curve  at  n^^.  As,  moreover,  we  su})- 
posc  the  ]>oints  ff^^  of  order  vi  to  be  everywhere  dense  upon  tiie 
curve,  it  must  be  a  natural  boundary. 

ft  is  a])parent  now  that  the  ex[)ression  ((>)  continues  the  initial 
i'unetion  </)(^,^)  across  a  natural  boundary  into  other  regions  where 
it  defines  in  similar  manner  other  analytic  functions  with  natural 
boundaries.  But,  it  may  be  askel,  is  there  any  ])roper  sense  in 
which  these  analytic  functions  may  l)e  regarded  as  a  continuation 
of  one  another?     Just   here  Bore/  steps  in   and,  after  imposing 


-J' 

2-'" 

v^! 

(1  ' 

(1-   ' 

(t 

130  THE  BOSTON  COLLOQUIUM. 

further  coiulitions,  shows  that  when  the  function  defined  by  (6) 
within  some  one  of  tiie  curves  is  zero,  the  functions  defined  within 
the  other  curves  must  also  vanish.*      Take  m  =  1,  so  that 

(7)  cp(r^  =  ^     ^^"     ■ 

^   ^  ^  '  ■  z  —  a^ 

])%•  a  linear  transformation 

<(Z  +  /> 

""  ^  cz  -h<l 

any  interior  ])oint  of  one  curve  may  be  taken  as  the  origin  and 
any  interior  point  of  a  second  curve  may  be  transformed  simul- 
taneously into  the  point  at  infinity  without  changing  the  ciiaracter 
of  the  series  to  be  investigated.  Now  at  the  origin  the  successive 
coefficients  in  the  expansion  of  (f>(^z)  into  a  Taylor's  series  are  the 
negative  of 

(«) 

while  those  in  the  cx})ansion  for  z  —  x  are 

lUn-i-l  j)roves  that  when 

lim  i"  .1    =  0, 

n  —-  J. 

the  ciictricients  (!*)  must  vanish  if  those  given  in  (8)  do.  Any  one 
of  the  analytic  fimctions  under  discussion  is  therefore  completely 
detei'inincd  l)y  any  other,  the  expression  (7)  being  the  intermediary 
bv  which  we  [)ass  from  one  to  the  other. 

So  tar  a^  yd  ap])cars,  this  metiiod  of  continuing  an  analytic 
function  across  a  natural  boundary  is  of  vcrv  limited  a])j)licability. 
Its  -ignilicancc  lias  been  made  clearer  by  Uoi-cfi^  later  memoir  in 
the  .  1  rill  MiiljiciiKificd .  I  lei'c  t  he  rational  fractions  an^  of  a  less  highly 
specializiil  chai'aeter,  but  the  essential  nature  of  the  investigation 
can  still  be  exhibited  without  abandoning  the  exjtression  (fi).  Let 
j  .1      ^.^  "      ',  whei'e  II ^^  denotes  the  ;/th  term  ol' a  convergent  series 

•'  Ct'.   pp.  :;■_'   :;:;  of  hi^  tlir.-i-  in-  jip.  lil-'.is  ol'  his  'I'h''iiri,-  ilr.<  fni,r'iiin.<. 


DIVERGENT  SI:RIES  AND  (OXTINrED  FKACTK )NS.     131 

of  [)ositiv('  miiuhers.  A\'e  shall  suj)posc  that  the  poles  of  the  terms  of 
(())  are  everywhere  dense  over  a  large  portion  of  the  i)lane,  leaving, 
iKjwever,  at  least  one  area  free  from  poles,  so  that  there  shall  be  an 
analytie  function  to  continue,  though  even  this  is  not  necessary. 
Borel  proves  that  parallel  to  any  assigned  direction  there  will  be 
an  intinity  of  straight  lines,  everywhere  dense  throughout  the 
plane,  along  which  the  series  (())  will  converge  absolutely  and 
uniformlv.  The  function  defined  along  these  lines  is  therefore  a 
continuous  one. 

The  ])roof  of  this  result  is  short  and  simple.  Describe  about 
the  [)oles  r;^  as  centers  circles  which  have  successively  the  radii 
",  (/(  =  1,  2,  •  •  •).  If  there  is  any  point  which  lies  outside  all  of 
these  circles,  the  series  (6)  must  there  converge,  since  at  such  a 
point  the  absolute  value  of  the  ni\\  term  is 

.1  w'"-^ 

that  is,  less  than  the  7(th  term  of  a  convergent  series  of  positive 
numbers.  But  are  there  ])oints  outside  of  all  the  circles?  To 
settle  this  ([uestion,  take  any  straight  line  perpendicular  to  the 
assigned  direction  and  })roject  orthogonally  all  the  circles  upon  the 
line.  The  total  sum  of  all  the  projections,  -2^'^,,,  will  be  conver- 
gent. Moreover,  by  cutting  off  a  sufficient  number  of  terms  at 
the  beginning  of  (G),  the  sum  of  the  projections  may  be  made  less 
than  any  assigned  segment  ab  of  the  line.  Let  ^V  terms  be  cut 
off  f  )r  this  purpose.  Take  any  point  e  of  the  segment  which  does 
not  lie  upon  the  projection  of  any  circle  nor  coincide  with  the 
projection  of  one  of  the  first  A'  poles  of  (•>).  At  e  erect  a  })erpen- 
dicular  to  al).  This  will  be  a  line  parallel  to  the  assigned  direc- 
tion which  throughout  its  entire  extent  lies  without  all  the  circles, 
excej)ting  possibly  the  fii'st  A'.  Hence  the  series  (U)  will  con- 
verge ab-olutcly  and  uniformlv  along  the  line,  even  though  the 
line  lie  intinitesinially  close  to  some  sut  of  poles  in  the  system. 
Lastly,  because  nh  was  an  interval  of  arbitrary  length,  these  lines 
of  convergence  nuist  be  everywhere  dense  throughout  the  plane, 
ol)viouslv  f)rming  a  non-enumerable  aggreuate. 


132  THE  BOSTON  COLLOQUIUM. 

Since  the  series  is  uniformly  convergent,  it  can  be  integrated 
term  by  term.  Clearly  also  the  numerators  A.  in  (6)  can  be  so 
conditioned  that  the  term-by-term  derivative  of  (6)  shall  be 
uniformly  convergent.  Then  the  derivative  of  (f>{z)  is  coincident 
with  the  derivative  of  the  series.  It  is  even  possible  to  so  choose 
the  A.  that  the  series  will  be  unlimitedly  differentiable. 

I  may  add  that  in  any  region  of  the  plane  there  will  be  an 
infinite  or,  more  specifically,  a  non-enumerable  set  of  points, 
through  each  of  which  passes  an  infinite  number  of  lines  of  con- 
vergence. If  a  closed  curve  is  given  it  will  be  possil)le  to 
approximate  as  closely  as  desired  to  this  curve  by  a  rectilinear 
polygon,  along  whose  entire  length  the  series  converges  and  defines 
a  continuous  function.  Integration  around  such  a  polygon  gives 
for  the  value  of  the  integral  the  product  of  2/7r  into  the  sum  of 
the  residues  of  those  fractions  whose  poles  lie  in  the  interior  of 
the  polygon.  Finally,  if  we  take  for  axes  of  x  and  y  two  perpen- 
dicular lines  of  continuity  of  ^(^),  all  the  lines  of  uniform  continuity 
which  meet  at  their  intersection  will  give  a  common  value  for  4>'{z), 
and  the  real  and  imaginary  parts  of  (f>(z)  will  satisfy  Lap/oce's 
equation  : 

^  .,+  ^  ,  =  (}. 

ex  CI) 

Tlius  we  have  in  (^(z)  a  species  of  quasi-monogenic  function. 
One  question  Bore/  has  as  yet  found  himself  unable  to  resolve. 
If  (f)(^z)  =  0  along  a  finite  portion  of  any  line,  will  the  series  in 
consequence  vanish  identically?  If  this  question  be  answered  in 
the  affirmative,  the  analogy  with  an  ordinary  analytic  function 
will  !)('  still  more  comj)let(>. 

Let  us  now  return  to  the  case  in  which  two  or  more  fiiuetions 
with  natural  boundaries  are  defined  by  (7j.  Th(!  lines  of  eon- 
tinuity  just  described  form  an  infinitely  thick  mesh-work  along 
which  (/)(■:)  can  l)e  carried  continuously  from  the  one  analytic 
function  into  the  others.  Suj)j)ose  again  that  the  origin  is  not  a 
])oint  of  condensation  of  the    ])oles  a^^  so  that  (^(:t)  can  be  expanded 


DIVKKGENT  SERIES  AND  CONTINUED  EKAC'TIONS.     loo 

at  the  origin  into  a  Maclanrin's  scries  1('.z\  Xow  if  a  ray  is 
(li-awn  froni  the  origin  through  the  pole  a^^  and  the  portion  of"  the 
rav  between  a^^  and  oo  is  retained  as  a  cut,  the  ///th  term  of  (7)  can 
be  expanded  into  a  series  of  polynomials 


which  converges  over  the  plane  so  cut.     The  series  (7)  can  there- 
fore be  resolved  into  a  double  series 


and  this  expression  will  be  valid  on  an  infinity  of  rays  from  the  origin 
which  do  not  pass  through  any  of  the  poles.  Since,  moreover,  the 
poles  are  an  enumerable  set  of  points,  these  rays  Avill  be  infinitely 
dense  between  any  two  arguments  which  may  be  taken.  By  fur- 
ther conditioning  the  yl  ,  Bord  is  able  to  rearrange  the  terras  of 
the  double  series  so  as  to  form  a  series  of  polynomials  y^  ^..(z), 

n 

in  which 


«^)=-l,t:'^.(i> 


and  in  this  way  he  obtains  a  series  of  polynomials  which  is  con- 
vergent on  a  dense  set  of  rays  through  the  origin. 

It  also  appears  that  the  polynomial  series  2  Q^i^^  can  be  formed 
directly  from  2('.?y  without  the  intervention  of  (7).  When,  there- 
fore a  Maclanrin's  series  is  given  which  corresponds  to  such  an 
expression  (7)  as  is  now  under  discussion,  the  continuation  of  the 
function  can  be  made  along  the  above  set  of  rays.  Now  the  rays 
cut  any  curve  upon  \vhich  either  (7)  or  S(p  (,-i)  defines  a  continuous 
function  in  a  set  of  points  everywhere  dense.  The  value  of  the 
function  along  the  entire  curve  therefore  depends  only  upon  the 
coefficients  c. ;  /.  e.,  upon  the  value  of  the  function  and  its  deriva- 
tives at  the  origin.  It  is  shown,  moreover,  that  any  point  of  the 
plane  which  is  not  a  point  of  condensation  of  the  poles  a,_  may 


134  TirE  BOSTON  COLLOQUIUM. 

be  converted  l)v  transformation  of  axes  into  such  an  origin. 
Finally,  Borel  gives  a  case  in  which  the  poles  may  be  everywhere 
dense  over  the  entire  plane,  so  that  the  function  defined  by  (7)  is 
nowhere  analytic,  and  yet  its  value  is  determined  along  the  lines 
of  continuity  by  the  value  of  the  function  and  its  derivatives  at 
the  origin.  Here  then  is  a  class  of  non-analytic  functions  sharing  a 
most  fundamental  property  in  common  with  the  analytic  functions! 
Is  it  not  then  possible,  as  Borel  surmises,  that  there  is  a  wider 
theorv  of  functions,  similar  in  its  outlines  to  the  theory  of  ana- 
lytic fimctions  and  embracing  this  as  a  special  case?  If  so,  the  con- 
ceptions of  Weiersfrass  and  of  Meray  are  capable  of  generalization. 


Part  II.     Ox  AlpxErraic  Continued  Fractions. 

I^ecturp:  5.      I\t<Je^s    Table  of  Approximanis  (ind  its 
Apjtlkdtions. 

Both  historicallv  and  pntspectively  one  of  the  most  suggestive 
and  important  methods  of  investigating  divergent  power  series  is 
bv  the  instrumentality  of  algel)raic  continued  fractions.  It  is  for 
this  I'eason  that  I  liave  ventured  to  combine  in  a  single  course  of 
lectures  two  subjects  apparentlv  so  unrelated  as  divergent  series 
and  continued  fractions.  I  shall  not,  however,  confine  myself  to 
the  consideration  of  the  latter  subject  solely  with  reference  to  the 
theorv  of  divergent  series.  ft  is  rather  my  ])ur])ose  to  give  some 
account  of  the  present  status  of  the  theory  of  algel)raic  continued 
fractions.  At  the  close  of  th(>  next  lecture  a  bil)liography  of 
memoirs  connected  with  tlu^  subject  is  apjH'nded,  to  which  refer- 
ence is  mad(^  throughout  this  lecture  an<l  the  next  by  means  of 
ninnbers  enclosi'd  in  s(|uar(^  brackets. 

I)V  the  term  (ihjfhroie  contiinied  fraction  is  undei'stood,  in  dis- 
tinction from  a  continued  fraction  with  mnncrical  elements,  one 
in  which  the  elements — /.  r.,  the  jnirtial  innnerators  and  denomi- 
nators —  are  functions  of  a  single  variable  x  or  of  several  varia- 
l)]e^   [I'l,  <i.  p,  4].      All  hough  the  term  algebraic  does  not  seem  to 


DIVKIKJKNT  SKKIKS  AND  (ONTINrKD  FPvACTTOXS.     LiO 

nie  to  1)1'  fortunately  (;lioscn,  I  shall  novcrtliclcss  accc])!  it  and  use 
it  to  indicate  the  class  of  coiitiiiU('(]  fractions  which  it  is  ]ii-o|»osf'(l 
to  consider  here. 

The  first  foundations  of  a  theory  of  continued  fractions  were 
laid  l)v  Killer,  who  early  enij)loyed  thetn  [1,  r/]  to  derive  from  a 
given  power  series 

/;,  +  /.-,'•  +  l^y  +  ■  •  ■  (/.•„  +  0) 

a  continued  fraction  of  the  form 

1  ((,.r  (i,x 


(V 


A,  +  h..r  +  f/,  +  />„.}■  +  (I.,  + 


A  second  form,  also  introduced   by  J'Juler*   [40,^^]    is  the  more 

familiar  one 

a„       rt,i-       a.,v       (i.,r 

^-^  1  +    1    +    1    -f     1    +  ■  ■  ■' 

wlii(;h  was  later  used  by  C/c/.^'.s'x  [-"M]  in  his  celebrated  continued 
fraction  for  F(7.,  (3,  y,  .r)'F(a,  j3  +  1,7+  1,  •'')•  From  this  time 
on  still  other  forms  were  discovered  so  that  it  became  impossible 
to  speak  of  a  unique  development  of  a  function  into  a  continued 
fraction.  Among  these  forms  may  be  especially  mentioned  the 
continued  fraction 

1  1  1 

C3  )  - 

^     '  o^.r  +  ('>,  +  a.^.r  +  b.,  -(-  a^  +  ^>3  +       ' 

used  by  Heine,  Tchehi/chef]  and  (others  in  a})])roxiniating  to  series 
in  descending  ])owers  of  .r.  By  the  substitution  of  l/.r  for  .r  and 
a  simple  reduction  this  can  be  transformed,  after  the  omission  of  a 
factor  .r,  into 

(3)  ^  x^-  x\ 

",+  '>,■>•  +  "2  +  l>^r  +  a.^x  +  b.^  + 

The  reason   for  this  variety  t)f  form  and  for  the  occurrence,  in 


*  Pade  in  his  thesis  (p.  08)  traces  it  l)ack  to   Laml)ert  [2,  a]  and   Lagrange, 
but  Killer's  use  is  earlier  still. 


136  THE  BOSTON  COLLOQUH'^I- 

|)artioulai',  of  tlie  three  types  just  given  is  discussed  by  Pade  in 
his  thesis  [IfJ,  ^r].  As  this  thesis  is  the  foundation  I'or  u  systematic 
study  of  continued  fractions,  it  will  be  necessary  to  give  a  recapit- 
ulation of  its  chief  results. 

Let 
(4)  ,S(.r)  =  c,^  +  c,r  +  e,r^  +  . .  .  (c,,  =  1) 

be  any  given  p(nver  series,  whether  convergent  or  divergent.  If 
iV  (.r)  7>  (,/■)  denotes  an  arbitrary  rational  fraction  in  which  the 
numerator  and  denominator  are  of  the  ^>th  and  qi\\  degrees  respec- 
tively, there  will  be  j>  -\-  <j  +  1  parameters  which  can  be  made  to 
satisfy  an  equal  number  of  conditions.  Ix't  them  })e  so  determined 
that  the  expansion  of  X  D^  in  ascending  powers  of  x  shall  agree 
with  (4)  ibr  as  great  a  number  of  terms  as  possible.  Iii  general, 
Ave  can  equate  to  zero  the  first  p  -\-  q  +  1  coefficients  of  the  expan- 
sion of  J)  6'(.r)  —  X ^  in  ascending  })owers  of  .r,  and  no  more. 
Hence,  unless  X  and  D  have  a  common  divisor,  the  series  for 
X  J>  agrees  with  (4)  for  an  equal  number  of  terms,  and  the 
approximation  is  said  to  be  of  the  (p  -f-  g  +  l)th  order.  In  excep- 
tional cases  the  order  of  the  ap])roximation  may  be  either  greater 
or  less.  J^dtle  examines  these  exceptional  cases  and  proves  strictly 
that  among  all  the  rational  fractions  in  which  the  degrees  of  numer- 
ator and  denominator  do  not  exceed  [>  and  (j  resj)cctively,  there 
is,  taken  in  its  lowest  terms,  one  and  only  one,  the  expansion  of 
which  in  a  series  will  agree  with  (4)  for  a  greater  number  of  terms 
than  any  othei'.  Such  a  rational  fraction  I  shall  term  an  approxi- 
mant  of  the  given  series. 

The  existence  of  a|)proximants  was,  of  course,  well  known 
bcfure  I'c'lr,  but  no  svstcmatic  examination  of  them  had  been 
made  exeej)t  bv  /•'/•oA(7//V.s  [  lo] ,  who  delei'inined  the  im])<>rtant 
I'clations  which  iKUMnallv  exist  between  them.  Padr  goes  further, 
and  arranges  the  apj)i'oxiinants,  expressed  each  in  its  lowest  terms, 
into  a  table  of  double  enti'v  : 


])rvi:K(;KXT  skkies  and  contixuei)  fkactioxs.    i;]7 


.\' 

-^;, 

7  =  " 

/>, 

—  '', 

'    />„, 

u 

7=1 

7  =  2 

U      '  1 


/A„, 


X, 


7 


When  the  order  of  approximation  of  a  rational  fraction,  taken 
in  its  lowest  terms,  is  exactly  equal  to  the  sum  of  the  degrees  of 
numerator  and  denominator,  increased  by  1,  the  fraction  will  be 
found  once  and  only  once  in  the  table.  If,  conversely,  a  fraction 
N  ID  occurs  but  once  in  the  table,  the  numerator  and  denomi- 
nator  are  of  degree  p  and  q  respectively,  and  the  order  of  the 
approximation  which  the  fraction  affords  is  exactly  ^5  +  7  +  1 . 
The  approximant  is  then  said  by  Fade  to  be  normal.  We  shall 
also  call  the  table  normal  when  it  consists  only  of  normal  fractions, 
or,  in  other  words,  when  no  approximant  occurs  more  than  once  in 
the  tal)le. 

Obviously  all  approximants  which  lie  upon  a  line  perpendicular 
to  the  principal  diagonal  of  the  table  correspond  >to  the  same 
value  of  y>»  +  7  -f  1.  Hence  in  a  normal  table  they  ap})roximate 
to  (4)  in  ecjual  degree,  and  accordingly  may  be  said  to  be  eqaal/i/ 
advanced  in  the  table.  If  y>  +  7  +  1  increases  in  passing  from 
one  fraction  to  another,  the  latter  is  the  more  advanced. 

Two  approximants  will  be  called  confifjuoux  if  the  squares  of 
the  table  in  which  thcv  are  contained  have  either  an  edtre  or  a 
vertex  in  common. 

Consider  now  a  normal  table,  and  take  any  succession  of  aj>prox- 
imants,  beginning  with  one  upon  the  l)order  of  the  table  and  pass- 
ing always  from  one  approximant  to  another  which  is  contiguous 
to  it  l)ut  more  advanced.  l\ide  shows  that  any  such  sequence  of 
approximants  makes  a  continued  fraction  of  which  the  approxi- 


138  THE  BOSTOX  C'OLLOQT'II'M. 

mants  are  the  sueeessive  convergents.  *  Thus  a  countless  manifold 
of  continued  fractions  can  l)e  formed,  any  one  of  which  through 
its  convergents  gives  the  initial  series  to  any  re(|uired  number  of 
terms  and  hence  defines  the  series  and  tahle  uniquely,  lu  all  of 
PdiJv'a  continued  fractions  the  j)artial  numerators  are  monomials 
in  X. 

The  continued  fraction  is  called  regular  when  its  partial  numer- 
ators are  all  of  the  same  degree  and  likewise  its  denominators, 
certain  specified  ii'regularities  l)eing  admitted  in  the  first  one  or 
two  partial  fractions.  These  irregularities  disappear  when  the 
continued  fraction,  as  is  most  usual,  commences  with  the  corner 
element  of  the  table.     (Cf.  the  continued  fractions  (2)  and  (o).) 

In  a  normal  table  a  regular  continued  fraction  can  be  ol)tained 
in  any  one  of  three  ways.  If  we  take  for  the  convergents  the 
approximants  which  fill  a  horizontal  or  vertical  line,  a  continued 
fraction  is  obtained  which  —  except  for  the  irregularity  permitted 
at  the  outset  —  is  of  the  form  (1)  given  above.  If  the  approxi- 
mants lie  upon  the  principal  diagonal  or  any  parallel  line,  the  con- 
tinued fraction  is  of  type  (•']).  Lastly,  if  the  convergents  lie  upon 
a  stair-like  line,  proceeding  alternately  one  term  horizontally  to 
the  right  and  one  term  vertically  downward,  the  continued  fraction 
is  of  the  familiar  form  (2). 

M'hen  a  table  is  not  normal,  the  a]>proximants  which  are  iden- 
tical with  one  another  are  shown  by  Padf  to  fill  always  a  square, 
the  edges  of  which  are  ])arallel  to  the  borders  of  the  table.  When 
the  sfpiare  contains  (n  -f  1)"  elements,  the  irregularity  may  be  said 
to  be  of  the  ;/th  or<ler.  The  vertical,  horizontal,  diagonal  and 
stair-like  lines  give  regular  continued  fractions  as  before,  uidess 
they  cut  into  one  or  more  of  these  s(piare  blocks  of  (Mpial  api)roxi- 
mants.  When  this  haj)])ens,  certain  irregularities  aj)pear  in  tiie 
continued  fraction  which  give  I'ise  to  vai'ious  difliculties  in  the 
consideration  of  matters  of  convergence  and  other  (piestions. 

( )n  {\\\<  aecoiitit  it  i>  natural  to  in(|uii'e  first  whether  the  con- 
tinue*] fraction  ha>  or  has  not  a  normal  chai'acter.      If  it   has,  the 

*  This  i^  ;il>~ii   t:icitl\'  iin]ilit.'i|    in  the   relation^  iri veil    l>y  l''r()ltcniii^  [].'!,   p.  •")]. 


DlVEKCiKNT  SF.IUKS  AM)  COXTIXUED  FRACTIONS,     l-*'!' 

existence  of  tlie  tliree  reuular  types  of"  continued  fractions  is  as- 
sured. The  neeessarv  and  suflicient  condition  that  the  taMc  shall 
be  normal  is  that  no  one  of  the  determinants 


a-/3 
i~  8- 


r_.  =  0  if  /  <  0) 


"a  '  a-  1  orS-l 

shall  vanish  [K)^  r/,  p.  oo].  It  will  be  noticed  that  the  determi- 
nants are  of  the  same  sort  as  those  which  play  S(^  conspicuous  a 
role  in  JladauKird's  discussion  of  series  representing  functions 
with  polar  singularities. 

So  far  as  I  am  aware,  the  normal  character  of  the  table  has 
been  established  as  yet  only  in  the  following  cases  :  (1)  for  the 
exponential  series  [37]  and  for  (1  +  :»•)'"  when  m  is  not  an  integer 
[35,  r/]  j|  by  Fade;  and  (2)  for  the  series  of  SV/e/Z/cs,  by  myself  [45]  . 

The  construction  of  P<:///(''.s  table  leads  at  once  to  a  number  of  new 
and  important  questions.  The  numerators  and  the  denominators  of 
the  apj)roximants  constitute  grouj)s  of  polynomials  which  it  is  only 
natural  to  expect  will  be  characterized  by  common  or  kindred 
properties.  The  table  then  affords  a  suitable  basis  for  the  classifi- 
cation of  polynomials.      Thus,  for  exam])le,  the  polynomials  of 

t  At  least  half  of  tlie  table  fori^(«,  1,  >,  /)  lias  a  normal  character.  This  was 
proved  incidentally  in  my  thesis  [70]  by  showing  that  the  remainders  corre- 
sponding to  a[)proxirnants  on  or  above  the  diagonal  of  the  table  were  all  distinct. 
The  method  of  conformal  re])resentation  was  there  employed,  but  the  same  fact 
can  also  be  demonstrated  very  simply  by  means  of  (iauss'  reldtiones  inter  contiguas 
(formulas  (19)  and  (20)  of  [34]).  Tiie  approximants  in  the  other  half  of  my 
table  f('f.  [Tfi],  p.  4-1)  were  constructed  on  different  ])rinciples  from  Pade'p, 
the  approximation  being  made  sinuiltaneously  with  reference  to  tva  points, 
r  =:  0  and  r=oc,  but  the  resulting  continued  fractions  were  of  the  same  form 
as  Pade's.  It  is  noteworthy  that  the  rehttioyiff  inter  contigiut.-i  lead  to  such  a 
table  rather  than  to  tlie  one  of  Pade's  construction. 

In  the  case  of  F{ —  m,  1,1,  — z)  =  (1  —  .')'"  the  half  of  Pade's  table  below  the 
diagonal  is  also  normal,  since  the  reci[)rocal  of  the  apjtroximants  in  the  lower 
half  are  the  approximants  in  the  upper  half  of  the  table  for 

Fim,l,l,-o:)  =  {l+x)-'\ 
The  normal  character  of  t  lie  table  for  C  then  follows  since  e^'^lim  F(fj,  1,1,  -^f/)- 


140  THE  BOSTON  COLLOQUIUM. 

Lcf/cii'h-c  aiul  similar  polynomials  are  obtained  from  the  series  for 
log  (1  —  •^')/(-^  +  •'')?  ^vhile  the  numerators  and  denominators  of  the 
approximants  for  (1  +.7-)'"  are  the  hypergeometric  jwlynomials 
/•'(  —  fx,^  —  V  ±  rn,  —  fJ^  -\-  v,  —  .r),  in  which  /x  and  v  arc  integers,  or 
the  so-called  ]Kdynomials  of  Joco/j!  [Go]  .  In  these,  as  in  numerons 
other  cases,  the  denominators  of  the  convergcnts  and  the  remainder- 
functions,'^'  formed  by  multiplying  each  denominator  into  the  cor- 
responding remainder,  are  solutions  of  homogeneous  linear  dif- 
ferential equations  of  the  2nd  order  which  have  a  common  group, 
and  the  relations  of  recurrence  between  three  successive  denomi- 
nators or  remainder-functions  are  the  relationei^  inter  contUpmH  of 
(kin.ss  and  IiifiiKoni.  (See  in  particular,  [75,  </]  and  [~')].) 
The  furthei-  study  of  such  groups  of  polynomials  will  probably 
bring  to  light  new  and  important  properties.  The  position  of 
the  roots  of  the  denominators  should  especially  be  ascertained,  be- 
cause tlie  distribution  of  these  roots  has  an  intimate  connection  with 
the  form  of  the  region  of  convergence  of  the  continued  fraction 
and  oftentimes  also  with  the  position  and  character  of  the  function 
which  the  continued  fraction  defines. 

Probably  the  most  fundamental  question  concerning  iV/c/t^'.s 
table  is  that  of  the  convergence  of  the  various  classes  of  continued 
fractions  or  lines  of  aji])roximants.  The  first  investigation  of  the 
convergence  of  an  algebraic  continued  fraction  was  made  by  Rie- 
iixiui)  [18]  in  18()3,  followed  by  Thome  [ID]  a  few  years  later.t 
I)oth  writers  investigated  the  continued  fraction  of  (iauss  by 
rather  j)ainful  methods,  not  based  al)Solutely  upon  the  algo- 
I'itlim  of  the  contimied  fraction  but  uj)on  extraneous  considera- 
tions. This  is  not  surpi-ising,  for  there  were  at  that  time  no  gen- 
eral criteria  for  the  convergence  of  continued  fractions  with 
<'(>iiijJ(.r  elements,  and  even  now  the  number  is  astonishingly 
small. 


'  III  lit  lc;i-t  lialf  of  tlio  tnl)lt".      See  the  preceding:  footnote. 

"f  Ah  Ivicinaiurs  work  appeared  postluiinoiisly,  TlioiiuVs  lia.s  the  priority  of 
publication  f  ls(;(i)  l)ut  was  itself  preceded  l)y  Worpitzky'fl  dissertation,  to  which 
reference  i.s  made  in  a  subsequent  footnote. 


D]VKrv(;KNT  SEKIKS  AXl)  CONTIXrKD  FRACTIONS.     141 

The  two   |)rin('ij)al   criteria   for  convergence   correspond    to  the 
familiar  tests  for  the  convergence  of  a  real  continned  fraction 

(-\  ^  ^-  ^3 

^''^  \   +   \  +  \   +    "'' 

in  which  either  (1)  all  the  elements  are  positive  or  (2)  the  ])artial 
denominators  X.  are  ])ositive  and  the  partial  numerators  ix.  are 
negative.  The  latter  class  of  real  continned  fractions  is  known  to 
converge  if  X.  =  1  — fx..  Fiuiu/slwim  [20]  has  shown  that  when 
the  elements  are  complex,  the  condition  |  X.  j  =  1  +  |  /u.  |  is  still 
sufficient  for  convergence.  If,  furthermore,  the  continued  frac- 
tion has  the  customary  normal  form  in  which  A*-,,  =  1 ,  the  condition 
may  be  replaced  In*  the  less  restrictive  one  [29,  p.  320], 

1  1    ■ 

The  necessary  and  sufficient  condition  for  the  convergence  of  the 
first  class  of  real  continued  fractions  can  be  most  easily  expressed 
after  it  has  been  reduced  to  the  form 

111  ,  , 

x;+x:  +  x;  +  ---  .      (^>'^)- 

If  then  2X'_  is  divergent,  the  continued  fraction  converges,  while 
it  diverges  if  2X'  is  convergent.*  But  in  the  latter  case  limits  exist 
for  the  even  and  the  odd  convergents  when  considered  separately. 
This  result  is  included  in  the  following  theorem  which  I  gave  in 
the  TranHdctloiiH  of  1901  for  continued  fractions  with  complex 
elements  [ol  ]  :    If  in 

1  1  1 

7,  +  //i?,  +  %,  +  ifi,  4-  ^3  +  //3.,  -f-  "  ' 

the  elements  a.  have  all  the  same  sitjn  and  the  3.  are  alternately 
positive  and  negative,  t   the  continued   fraction  will   converge  if 

So;-!-  //3^  I  is  divergent  ;  on  the  other  hand,  if  S    a^  +  //3^^  ;    is 

"•■'Seidtjl,  IInl)ili/nli„ii.<Hrlirljl,  ISIH,  and  Stern,  Jonrn.  J'ilr  Mdfli.,  vol.  :!7  (1848), 
[).  2t;'.t. 

i"  Zero  values  are  permissible  Air  (.'itlier  ",   or  .^. 


142  THE  J30ST0X  COLLOQUIUM. 

fonvtM-^cnt  and  e'ltlicr  tlio  n..  or  the  /3.  fulfill  the  condition  just 
.stated  concerning  their  signs,  the  even  and  the  odd  convergents 
iiave  se[)arate  limits. 

The  most  general  criterion  for  the  convergence  of 

K        !>■:        K 

1    +  1    +  1    -f  ■■■ 

(6^  real  or  complex)  seems  to  l)e  the  one  which  I  gave  in  October, 
1901   [32, />,  §5]. 

Two  remarks  of  a  general  nature  concerning  the  convergence 
of  algebraic  continued  fractions  may  be  of  interest.  In  the  con- 
sideration of  numerical  continued  fractions  a  difficulty  frequently 
encountered  is  that  the  removal  of  a  finite  number  of  partial 
fractions  ^t./A..  at  the  beginning  of  (5)  may  affect  its  convergence 
or  divergence.  The  convergence  is  therefore  not  determined 
solely  by  the  ultimate  character  of  the  continued  fraction,  as  is 
true  of  a  series.  Pr'n\(/Hhchn  [29]  has  ])r()posed  to  call  the  con- 
vergence unconditional  when  it  is  not  destr(n-ed  by  the  removal 
of  the  first  /;  ])artial  fractions  of  (o).  The  difficulties  due  to  con- 
ditional convergence  usually  disappear  from  consideration  in  treat- 
ing algebraic  continued  fractions.  For  let  X^JD^  now  denote 
the  //th  convergent.  If  after  the  removal  of  the  first  ii  partial 
fractions  the  contimied  i'raction  converges  uniformly  in  a  given 
region  and  accordiiiglv  represents  a  fimction  F{z)  which  is  holo- 
morj)liic  within  tlie  I'cgion,  then  after  the  restoration  of  the  initial 
term>  the  ci)ntinuc(l  traction  will  define  the  I'unction 

which  inu>t  be  cither  holoiiiorphic  or  mcroinorpliic  within  the  given 
region  .■;2, '/ oi' c]  .  An  exception  occurs  onl\'  it' the  denominator 
of  (t'l)  \'ani-hes  idcnticalK'  in  the  region,  'i'his  is  iinpo.-'-ibh'  for 
the-ccoiid  and  third  tvpes  ol'  contiiUKMl  fi'actioiis,  >ince  the  de- 
velopnirnt  of  ;i  rational  iVaction  —  I)  I)  ^  in  cithci-  tv|)c  (2) 
Ol'  {'■'>)  con-i-t-  ot'  a  tinite  number  ot"  terms,  wlici'cas  the  develop- 
niciii  oj'  /•'-.),  1)\'  hvpoihe^is,  continues  indelinitely. 


i)TVI-;r(;i:xt  series  and  coxtixued  ekactions.    1^3 

The  second  remark  relatino:  to  conver<rcnce  is  that  its  discus- 
sioii  lor  a  contiimed  frac^tion  is  usually  reduced  to  the  correspond- 
in<r  ((ucstion  for  an  inlinite  scries.     The  succession  of  convcrgents 

J)  J     />„.,/     X>„.^' 
is,  in  fact,  obviously  ecjuivalent  to  the  series 

/;+(/r;-7;)  +  (i"':-y;::)+"- 

n  \        11^1  11/  \        H^  2  (l-rl/ 

But  the  latter  by  means  of  the  familiar  relations  connecting  the 
denominators  or  the  numerators  of  three  consecutive  convergents 
nuiv  be  reduced  to  the  form  : 


u  \         n       ivTi  /i-rl 

(■) 


n-rz       (i-rJ 

A\'e  turn  now  from  these  general  considerations  to  the  questions 
of  convergence  connected  ^vith  Pade^s  table.  Under  what  con- 
ditions will  the  various  lines  of  ap})roximants  converge;  in  par- 
ticulai",  the  three  standard  types  of  continued  fractions  obtained 
by  folhtwing  (1)  the  horizontal  or  vertical  lines,  (2)  the  stair-like 
lines,  and  (o)  the  diagonal  lines?  AVhen  they  converge  simid- 
taneously,  have  they  a  common  limit?  If  not,  what  are  the 
mutual  I'elations  between  the  functions  which  they  define?  What 
is  the  form  of  the  region  of  convergence  ? 

These  and  other  (juestions  press  upon  us,  and  are  of  great  in- 
terest. A  comj)lete  investigation  has  been  made  only  for  the 
exj>onential  series.  I'ddf  [-'T,  rr]  finds  that  when  p  7  for  any  suc- 
cession of  a])i)roximants  A'  J)  conver^-es  to  a  value  «,  the  ap- 
proximants  converge  toward  the  generating  function  r'  for  all 
values  ot'  ,/■.  Fui'tluM'more,  the  numerators  and  denominators  sepa- 
rately converge,  the  former  to  the  limit  (•'"■' "' '  ',  the  latter  to  r~-'' '"  '. 
"^riiis  -niootli  result  is  not,  however,  a  tyj)i('al  one,  not  even  for 
entire  functions.      ft  is  due  at    least  in    part   to  the  fact   that  r-"'  is 


144  THE  1K)ST()X  C()LL0(2Un'-^r. 

an  entire  function  without  zeros.  Tliis  will  be  apparent  after  an 
examination  has  heen  made  of  the  vertical  and  horizontal  lines 
of  Pd'lei!  table,  which  we  now  ])roceed  to  consider. 

It  is  ol)vious  that  the  first  p  +  q  +  \  terms  of  the  given  series 
(4)  determine  an  equal  number  of  terms  of  the  series  for  its  re- 
ciprocal. If,  therefore,  in  the  table  eacii  ap})roximant  is  replaced 
by  its  reciprocal  and  the  rows  and  columns  arc  then  interchanged, 
we  shall  obtain  the  table  for  the  reciprocal  series.  The  problems 
])resented  by  the  horizontal  and  vertical  lines  of  the  table  are  con- 
se(}uently  of  essentially  the  same  character,  and  our  attention  may 
be  confined  henceforth  to  the  horizontal  lines  alone. 

Hy  the  interchange  just  described  the  zeros  and  })()les  of  (4) 
become  the  j)oles  and  zeros  respectively  of  the  reci})roeal  function. 
In  the  case  of  the  exponential  function  the  reciprocal  series  has 
the  same  character  as  the  initial  series,  each  defining  an  entire 
function  without  zeros,  and  the  simultaneous  convergence  of  rows 
and  columns  for  all  values  of.''  was  therefore  to  be  ex])ected  ;  but 
in  general  this  does  not  hold. 

In  investigating  the  convergence  of  the  horizontal  lines  the  first 
case  to  be  considered  is  naturally  that  of  a  function  having  a  number 
of  poles  and  no  other  singularities  within  a  ])rescrii)e(l  distance  of 
the  origin.  It  is  just  this  case  that  Mont('HH]{H  [.'>.'),  '/]  has  exam- 
ined very  recently.  Some  of  you  Jiiay  recall  that  fotu'  years  ago  in 
th(,'  Cambridge  colloquium  Professor  Osr/ootl '■  took  Ilailtn/Kird's 
thesis  T  as  the.'  basis  of  one  of  his  lectures.  This  iKnable  tiiesis  is 
devoted  chielly  to  series  detlning  functions  with  jiolar  >ingu]arities. 
,}foiifissi/s  l)uilds  upon  this  thesis  and  a]>plies  it  to  a  table  j)ossess- 
ing  a  noi'mal  character.  Although  his  j)roof  is  subject  to  this 
limitation,  his  conclusion  is  nevertheless  valid  \vhen  the  table  is 
not  normal,  as   I  <liall  >liow  in  some  subse(|Ueiit  paper. 

The  flist  lioi-izdiital  row  of  the  table  scarceK'  needs  considera- 
tion, i'ny  it  consist-  of  the  polvnoinials  dbtained  bv  taking  suc- 
ee<.-ivel\-  1,  '2,  '.),  ■■■  tenn<  of  the  serie-.  ( "onse<|nenti  v  the  con- 
tiiMied  iVaction  obtained  iVoin  the  first   row, 

■  l!;ii.  ■■//!,.    Annr.   Mm  I,,  x,,,..,  vol.  :,,  I,|..  71-7S. 

i  ./ I'.    '/.     .1/.;//,.,    .,T.     1,    V..1.    >    I   \>'X1.. 


DIVERGENT  8ERIES  AND  COXTIXUED  ERACTIOXS.     145 

1    —  (i^.r  +  f'„  —  «o-''  +  ^'i  —  a.^j'  +  a,,  —        ' 

is  identical   with  the  series,  and   its  region  of  convergence  is  a 
circle. 

Let  ii*,  be  the  radius  of  this  circle  and  q^  the  number  of  poles 
of  (4)  which  lie  upon  its  circumference.  Suppose  also  that  the  next 
group  of  poles,  q.,  in  number,  lie  upon  a  circle  of  radius  7?^,  hav- 
ing its  center  in  the  origin  ;  that  q.^  poles  lie  upon  the  next  circle 
(A'g) ;  and  so  on  indefinitely  or  until  a  circle  is  reached  which  con- 
tains a  non-polar  singularity.  Hadamard  {I.  c,  §  18)  has  proved 
that  the  denominators  /),,  of  the  approximants  of  the  {q^  +  l)th 
row,  of  the  (7^  +  7.,  -f  l)th  row,  and  so  on,  approach  a  limiting 
form  as  we  advance  in  the  row,  and  that  the  limiting  polynomials 
give  the  i)ositions  of  the  first  q^,  7,  +  q.,,  •  •  •  poles  respectively. 
Thus  if,  for  example, 

and 

the  first  group  of  poles  are  the  roots  of  the  polynomial 
1  -i- B^x -{■  ••■B^^i'''\  Using  this  result  of  lladamdrd,  Montes- 
siis  shows  that  in  a  normal  table  the  approximants  of  the  (7,H-l)th 
row  converge  at  every  point  within  the  circle  (A*^)  —  excepting, 
of  course,  at  the  7^  poles  —  but  not  without  this  circle  ;  that  the 
a])proximants  of  the  (7,  +  (/^  +  1)^^^  ^'o^^'  converge  similarly  within 
the  circle  (//.j)  except  at  the  included  q^^  +  q^  poles;  and  so  on. 
In  proving  this  JIo)de.%s-us  makes  use  of  an  idea  advanced  in 
J/'ade's  thesis  ([16,  a,  p.  ol]  ,  or  [24])  which,  though  a])plicable  in 
the  present  case,  is  possibly  somewhat  misleading.  In  Fades  con- 
tinued fractions  the  partial  numerators  ^.  are  monomials  in  .»■.  This 
is  due  to  the  fact  that  there  is  a  steady  increase  in  the  order  of  the 
approximation  afforded  by  the  successive  convergents  at  .v  =  0. 
Consider  now  the  series  (7),  and  let  .7' denote  the  region  or  set  of 
points  in  the  .7'-plane  for  which  j  I)^^  ',  from  and  after  some  value 
of  n,  has  both  an  upper  and  a  lower  limit.      Then   in   T  the  con- 


14(5  THE  BOSTON  COLLOQTIUM. 

tinned  fraction  will   converge  or  diverge  sinmltaneonsly  with  the 
power  series, 

Call  ('  the  circle  of  convergence  of  (8).  At  all  points  of  T 
within  ('  the  continned  fraction  converges,  and  at  all  exterior 
})oints  of  7'  it  diverges.  On  tliis  account  l\i(Ie  proposes  to  call 
C  the  ''  circle  of  convergence  "  of  the  continued  fraction.  In  the 
case  which  we  iiavc  just  been  discussing  tliis  conce})t  is  applicable 
because  of  the  existence  of  limiting  forms  for  the  denominators  of 
the  rows  considered.  The  region  7^  com])rises  the  entire  finite 
])lane  with  the  exception  of  the  roots  of  the  limiting  form,  and 
the  circle  ('is  successively  identical  with  (Ji.-,),  {f^.^),  ....  Thus, 
as  we  pass  down  tlie  rows  of  the  table,  we  obtain  continued  frac- 
tions having  an  increasing  region  of  convergence. 

In  introducing  the  term  circle  of  convergence  for  a  continued 
fraction  Padf  ignores  all  points  not  included  in  T.  Call  the  ex- 
cluded ])oint-set  7".  If  I  />  f  increases  indefinitely  with  increas- 
ing  /*  over  tiie  whole  or  a  part  of  T  the  series  (7)  may  converge, 
and  this  may  haj)])en  even  though  (<S)  is  a  divergent  series.*  The 
term  circle  of  convergence  is  therefore  an  inappropriate  one,  al- 
though the  considerations  upon  which  it  is  l)ased  are  useful. 

Nothing  more  of  account  seems  to  be  known  concerning  the 
the  convergence  of  the  lioi'izontal  and  vertical  lines. t  The  more 
common  and  irnj)oi'tant  continued  fractions  are  obtained  from 
diagonal  and  stair-like  paths  throngh  tlie  table.  In  many  familiar 
contimied  fractions  of  the  second  tvpe, 

/.n  "o        "r>-       ".■'•       V 

^~'  I    -f     1     +     1     +    1     +■'■' 

*'l"lic  cnftlicii'iits  ill  tlir  (■iiiitiiiiicii  fnictioii  (if  Stieltjcs  ( (lisc'ii>>c(l  Icitrr  in  tlie 
lecture  i  can  lie  ca^ilv  sd  deterniincil  as  to  ;,Mve  a  case  nf  this  sort,  the  rej^'ioii  of 
coiivcrLrrncc  of  (7)  hciiiL,'  the  entire  i>hme  with  the  exceiitioii  of  tlie  tiet^ative 
half  of  the  real  axis.  We  siippo-c.  with  Padi'  tliat  the  ahsoliilc  tenii  of  /^,  is 
taken  i<|ual    to  ]  . 

i  it  i^  |Mrlia|iS  woilh  noting  that  the  eofllicieiits  in  the  lirst  type  of  coiitiiilK'tl 
frai'tioii-,  can  not  he  sclrcteil  a  rhi  tr.arily  if  it  i-'  to  he  coimeeted  with  such  a  tahle 
;u-~  I'adi' coii'-triK-ts.       in  the  other  two  Ivpcs  the  cocfiieients  are  eiiti  rcl  v  arhitrarv. 


I)1VKR(;KXT  SKRIES  AXI)  COXTINIIID  FRACTIONS.     147 

(1^^  with  increasing  n  a])pr()aclies  a  limit,  as  for  instance  in  the  con- 
tinued fraction  of  Gauss'  where  liin  c^  =  —  ].  Tiie  significance  of 
the  existence  of  such  a  limit  I  first  pointed  out  for  a  comprehen- 
sive class  of  cases  in  1901  [.'32,  (('],  and  since  then  I  have  shown 
by  simj)ler  methods  [32,  c]  that  the  result  is  })erfectly  general. 
liCt  lim  a  =  /•.  Then  the  continued  fraction  converges,  save  at 
isolated  ])oints,  over  the  entire  plane  of  x  with  the  exception  of 
the  wliole  or  a  jiart  of  a  cut  drawn  from  x  =  —  1  f4k  to  .r  =  oc  in 
a  direction  which  is  a  continuation  of  the  vector  from  x  =  0  to 
X  =  —  1 1 4k.  Within  the  plane  thus  cut  the  limit  of  the  continued 
fraction  is  holomorphic  except  at  the  isolated  points  which  (if  they 
exist)  are  poles.  When  there  is  no  limit  for  a^  but  only  an  upper 
limit  ("  for  its  modulus,  the  continued  fraction  (see  [32,  6])  is  mero- 
morphic  or  holomorphic  at  least  within  a  circle  of  radius  IjAU 
having  its  center  in  the  origin.*  A  special  case  is  that  in  which 
lim  (i^^  =  0.  The  limit  of  the  continued  fraction  is  then  a  function 
which  is  holomorphic  or  meromorphic  over  the  entire  plane.  A 
comparison  of  this  last  result  with  that  of  }Tordesms  shows  that  a 
much  greater  region  of  convergence  has  now  been  obtained.  This 
is  doubtless,  in  general,  a  reason  for  preferring  the  second  and 
third  types  of  continued  fractions  to  the  first. 

As  another  illustration  of  the  second  type  of  continued  fraction 
I  shall  choose  the  celebrated  continued  fraction  of  tStieUjei^  [2G,  «]. 
In  this  each  coefficient  a^^  is  ])Ositive.  By  putting  x  =  l/z  in 
(2),  the  continued  fraction,  after  dropping  a  factor  z,  can  be  thrown 
into  the  form 

(S')  ■••,  (''  >0), 

which   is  the   form    preferred   by   Stielfje.-^.     To   every   such   con- 
tinued fraction  there  corresponds  a  series 

■■■  A  (leiiionstration  of  tliis  j)r()perty  witliin  tlie  circle  (l/4f')  has  been  pre- 
viously ;riveii  ill  a  dissertation  liy  WorpitzLij  [IS  bis],  which  has  come  to  my 
notice  for  the  lirst  time  during  the  examination  of  tlie  proof-sheets  of  these  lec- 
tures. Tliis  dissertation  liears  the  date  ISO")  and  appears  to  he  the  earliest  pub- 
lislied  memoir  treatino;  of  tlie  convergence  of  algebraic  continued  fractions. 


148 


THE  BOSTON  COLLOQUIUM. 


c,.      c,       e„      c. 


(9) 

_0 

_   '-1 

+  .'- 

.'+••• 

for  which 

^0 

^1 

^'2           • 

••     ^-l      1 

A= 

^1 

c. 

^3           • 

j 

c^_ 

1    «n 

^.+1 

r 
■  ■   *^2n-2  !i 

(10) 

^1 

^2 

^3           • 

'  '     ^a 

J\  = 

^2 

^3 

C4           • 

■   •     C,H-1 

';l+l         /i  +  2 


The  correspondence  is  also  a  reciprocal  one.  To  every  series 
which  fulfills  these  conditions  there  corresponds  a  continued  frac- 
tion of  the  above  type  with  positive  coefficients.  From  the  con- 
ditions nO)   it  follows  that  c.>0  and  that  e  /e    ,>c    ,/c    ,, 

\  /  t  ^  nl      n  —  1  -^        n—ll      n  —  2 

If,  therefore,  the  increasing  ratio  c,Jc^^_^  lias  a  finite  limit,  the 
series  is  convergent.  On  the  other  hand,  if  it  increases  without 
limit,  the  series  is  divergent. 

In  investigating  the  convergence  of  the  continued  fraction  the 
especial  skill  of  Stieltjes  was  shown.  From  the  relation  connect- 
ing three  consecutive  denominators  (numerators)  of  the  conver- 
gents  it  was  shown  easily  that  either  set  of  alternate  denominators 
(numerators)  made  a  Sturm's  series,  whence  it  follows  that  all  the 
roots  of  the  denominators  (numerators)  lie  upon  the  negative  half 
of  the  real  axis  of  z.  This  leads  naturally  to  the  conjecture  that 
the  region  of  convergence  will  be  the  entire  j)lane  of  ~  with  the 
exception  of  the  whole  or  a  part  of  the  negative  half  axis,  and 
that  the  functional  limit  will  have  no  zeros  exterior  to  this  half 
of  the  axis.  First  the  convergence  is  examined  when  ~  is  real 
and  j)ositive.  The  criterion  of  Sriifr/^  cit(>d  previouslv  in  this 
lecture,  then  ai)j)rK's.  If,  namely,  ^^/'  is  divergent,  the  continued 
fraction  will  converge  along  the  ]K)sitive  axis,  while  if  ^'^',',  is  con- 


DlVKlKiKXT  SERIES  AND  CONTINUED  FKACTIONS.     149 

vergent,  the  two  sets  of  alternate  coiivergents  have  limits  which 
are  distinct.  The  conclusion  is  next  extended  by  Stielfjcs  to  the 
half  of  the  complex  plane  for  which  the  real  })art  of  z  is  positive. 

This  brings  him  to  the  difficult  part  of  his  problem,  the  exten- 
sion of  the  result  to  the  other  half-])lane  but  with  exclusion  of  the 
real  axis.  Here,  particularly,  Sfielfje.'^  [20,  a,  §  30]  shows  his 
ingenuity.  He  overcomes  the  difficulty  by  establishing  first  a 
preliminary  theorem  which  is  of  vital  importance  for  sequences 
of  polynomials  or  rational  fractions.  The  theorem  is  as  follows. 
Let  J\{^-),  f./^^),  ■  ■  •  be  a  sequence  of  functions  which  are  holomor- 
phic  within  a  given  region  T,  and  supj)ose  that  S,''^i/^^(2)  is  uni- 
formly convergent  iu  some  })art  T"  of  the  interior  of  T.  Then 
if  _/',(c)  -i-f.('i)  +  ■  •  •  +/„(''^)  ^^^^•''  '^"  upper  limit  independent  of  n  in 
any  arbitrary  region  T'  which  includes  T"  but  is  contained  in  the 
interior  of  7',  the  series  ^/^(s:)  will  converge  uniformly  in  T'  and 
therefore  has  as  its  limit  a  function  which  is  holomorphic  over  the 
whole  interior  of  T^ . 

In  the  application  of  this  theorem  /Stielfjcs  discomposes  each 
convergent  X^^{z) / DJz)  int(j  partial  fractions, 

JL  M,  M 

1  -^ -  -f Y 

z  +  a,       z  +  a.,  z  -^  a^ 


in  winch 


i  =  l 

From  this  it  follows  that  X^JI)^^  has  an  upper  limit  independent 
of  /(  in  any  closed  region  of  the  plane  which  does  not  contain  a 
})oint  of  the  negative  half-axis.  If  now  in  either  the  sequence 
of  the  odd  convergents  or  of  the  even  convergents  we  denote  the 
nth  term  of  the  sequence  by  X //)^  and  place 

the  series  ^^^^^fjj-:^  converges  uniformly  in  any  portion  of  the  plane 

*  For  a  further  extension  of  this  line  of  work,  see  Osgood,  Ayinah  of  Math., 
ser.  2,  vol.  :;  (IHUI),  p.  25. 


loO  THE  BOSTON  COLLOQUIUM. 

for  which  the  real  part  of  z  is  })ositive.  All  the  conditions  of  the 
lemma  of  Sdcltjes  are  now  fulfilled,  and  the  region  of  convergence 
may  be  extended  over  the  entire  ])lane  with  the  exception  of  the 
negative  half-axis. 

On  account  of  the  uniform  character  of  the  convergence  the 
limit  of  either  sequence  is  holomorphic  at  every  point  exterior  to 
the  negative  half-axis.  When  2«,',  is  divergent,  the  two  limits 
coincide  and  the  continued  fraction  itself  is  convergent.  On  the 
other  hand,  if  2«,'j  is  convergent,  the  two  limits  are  distinct. 
SticKjes  shows  also  that  in  the  latter  case  the  numerators  and  the 
denominators  of  either  secpicnce  converge  to  holomorphic  functions 
y>(-j),  'j(z)  of  (jenre  0,  and  the  two  pairs  of  functions  are  connected 
bv  the  e(piation 

7(^)/>,(^)-7.(^)/>(^)=l, 

which  corresponds  to  the  familiar  relation 

A,  a;,_, -/>,„_,  A^„=i. 

A  more  direct  method  [31]  of  demonstrating  the  convergence 
results  of  Sfie/fjes  is  by  an  extension  *  of  the  criterion  previously 
cited  for  the  convergence  of  continued  fractions  in  which  the 
partial  fractions  1  /(a^^  -f-  //3  J  have  an  a^  of  constant  sign  and  a 
/3^^  of  alternating  sign.  The  introduction  of  the  lemma  of  Sficlfjes 
is  consequently  unnecessary,  but  I  wish  nevertheless  to  emphasize 
its  fundamental  importance.  Other  notable  results  which  it  will 
be  impossible  to  reproduce  here  are  also  contained  in  his  splendid 
memoii'. 

*  If  nanu'ly,  ii  |  n„  -\-  iS^  \  is  divciM^ent  and  tlu'  ('(iiKlition  foiicernini^  tlie  signs 
citlicr  i)f  tlic  "„  or  of  tin*  ^,i  is  fuililied,  the  continiu'd  fraction  will  coiivt'ri,'-!.'  pro- 
vided |"„!/i  ^,1 1  lias  a  lower  or  an  upper  limit  respectively.  I'm  now  :  v^  in 
(>')  so  tliat  it  luM'onies 

'('  '  '  ■)■ 

"■  V  "i"'  ;■  ".'"■    '    "3"'  ;-       / 

When  ^"'  is  divergent,  this  I'alls  inider  the  extended  ei'iterion  if  we  put 
o'l/-  II  •  I  !  ,  except  when  ."  is  negative.  (  »n  the  otlu'r  hand,  when  ^"  is  con- 
vergent, the  criterion  applies  without  extension  directly  to  (^').  In  either  case 
the  iinil'orni  character  of  t  he  con  veruence  follows  with  the  addition  of  a  few  lines. 


DIVERGEXT  SEKIES  AND  CONTIXrED  ERACTIOXS.     l^")! 

It  is  interesting  to  bring  this  work  of  Sdc/fjcs  into  eonneetion 
with  the  table  of  P(i(lf  [44].  The  odd  convergents  of  the  con- 
tinued fraction  of  Sfic/tjes  fill  the  principal  diagonal  of  P<i(h'^s 
tal)le,  thus  constituting  i)y  tliemselves  a  continued  fraction  of  the 
third  type,  and  the  even  convergents  fill  the  parallel  file  immedi- 
ately below,  forming  a  similar  continued  fraction.  The  signifi- 
cance of  distinct  limits  for  the  two  sets  of  convergents  is  thus 
made  clearer. 

The  series  of  .Sfie/fjes  has  perhaps  its  greatest  interest  -when 
treated  in  connection  with  the  theory  of  divergent  series.  Although 
the  continued  fraction  always  converges  if  the  series  does,  the  con- 
verse is  not  true.  For  when  the  series  (9)  is  divergent,  two  cases 
are  to  be  distinguished  according  as  Id^  is  divergent  or  conver- 
gent. In  tiie  former  case  the  continued  fraction  gives  one  and 
only  one  functional  equivalent  of  the  divergent  series.  Le  Roy 
states,*  though  without  proof,  that  the  function  furnished  is 
identical  with  the  one  obtained  from  the  series  by  the  method  of 
J>orel,  whenever  the  latter  method  is  applicable  also..  When  2«/  is 
convergent,  two  different  functions  are  obtained  from  the  con- 
tinued fraction,  the  one  through  the  even  and  the  other  through 
the  odd  convergents.  And  if  there  are  two  such  functions  which 
correspond  to  the  series,  there  must  be  an  infinite  number.  For 
if  (^(.r)  and  ^{-i'),  when  expanded  formally,  give  rise  to  the  same 
divergent  series,  so  also  will 

in  which  c  denotes  an  arbitrary  constant.  Special  properties, 
however,  attach  themselves  to  the  two  functions  picked  out  by  the 
continued  fraction  of  Sficlfjfx,  upon  which  we  can  not  linger  here. 
This  result  of  'S7/(7/yV,s>  seems  to  me  to  be  especially  significant, 
since  it  indicates  a  division  of  divergent  series  into  at  least  two 
class(>s,  tho  one  class  containing  the  series  f  )r  which  there  is  prop- 
erly a  single  functional   ecjuivalent  and   the  othei-  com})rising  the 

*  Loc.  cit.,  p.  428. 


l-"32  THE  BOSTON  COLLOQUIUM. 

series  wliicli  correspond  to  sets  of  functions.  It  is,  of  course, 
just  possible  that  this  distinction  may  be  due  to  the  nature  of  the 
algorithm  employed  in  deriving  the  functional  equivalent  of  the 
series,  but  it  is  far  more  probable  that  the  difference  is  intrinsic 
and  independent  of  the  particular  algorithm.  If  this  view  be  cor- 
rect, the  method  o^  Borcl  which  gives  a  single  functional  equivalent, 
is  limited  in  its  application  to  series  of  the  first  class. 

An  extension  of  the  work  of  SHeUjcs  has  been  sought  in  two  dis- 
tinct directions  by  modification  of  the  conditions  imposed  upon  his 
series.  Bord  [4-3]  so  modifies  them  as  to  make  the  series  (when 
divergent)  fulfill  the  requirement  imposed  in  lecture  "2  and  permit  of 
manipulation  precisely  as  a  convergent  series.  In  the  last  number  of 
the  Transactions  *  [45]  I  began  a  study  of  series  which  are  subject  to 
only  one  of  the  two  restrictions  expressed  in  the  inequalities  (10), 
but  was  obliged  to  bring  the  work  to  a  hurried  close  to  prepare  these 
lectures.  In  the  main,  the  corresponding  continued  fractions  have 
the  same  ])roperties  as  the  continued  fraction  of  Stielfjcs,  but  a  con- 
siderable difl'erence  is  shown  in  regard  to  convergence.  Though 
the  roots  of  the  numerators  and  denominators  of  the  convergents 
are  still  real,  they  are  no  longer  confined  to  the  negative  half 
of  the  real  axis,  and  may  be  infinitely  thick  along  the  entire 
extent  of  the  axis.  In  certain  cases  the  continued  fraction  con- 
verges in  the  interior  of  the  positive  and  negative  half  planes, 
defining  in  each  an  analvtic  function  which  has  the  real  axis  as  a 
natural  i)()undary.  The  continued  fraction  therefore  effects  the 
continuation  of  an  analytic  function  across  such  a  boundary,  and 
gives  H  natural  instance  of  such  a  continuation  t  —  natural  in  dis- 
tinction from  artificial  cxam])lcs  set  up  with  the  ex]>ress  object  of 
showing  the  possibility  of  a  uni(}ue,  non-analytic  extension. 

Pa(h'  [1  7,  ii~\  has  suggested  the  foundation  of  a  theory  of  diver- 


-.Jiily.  \w:.. 

I  I'liirlicr    iiistniict's  of    ;i    natiinil   contiiniation  are  also  to  be  fouiul,   as,    for 
exainple,  lliat  alTonli'd  \iy 

acr(i>s  the  a\i^  ot  reals. 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     lo3 

gent  series  upon  the  continued  fractions  of  his  table.  The  diffi- 
culties of  carrying  out  the  suggestioii  are  undoul)tedly  very  great 
and  have  been  pointed  out  by  7>o/y'/.*  Xot  only  must  the  con- 
vergence of  the  principal  lines  of  approximants  and  tlie  agreement 
of  their  limits  be  investigated,  but  the  combination  of  two  or  more 
divergent  series  must  also  be  considered.  It  is  not  enough  to  point 
out,  as  does  Pade,  that  the  ap})roximants  of  given  order  for  any 
two  series,  whether  divergent  or  convergent,  determine  uniquely 
the  approximants  of  the  same  or  lower  order  for  the  sum-  and 
product-series.  For  practical  application  of  the  theory  it  must  be 
proved  also  that  the  function  defined  by  the  table  corresponding 
to  the  new  series  is,  under  suitable  limitations,  the  sum  or  product 
of  the  functions  defined  by  the  given  divergent  series.  But  great 
as  are  the  difficulties  of  such  an  investigation,  even  for  restricted 
classes  of  series,  the  reward  will  probably  be  corres})ondingly 
great. 

So  far  as  it  has  been  yet  investigated,  the  diagonal  type  of  con- 
tinued fractions  seems  to  have  accomplished  nearly  everthing  that 
can  fairly  be  asked  of  a  sequence  of  rational  fractions.  Not  only 
does  it  affijrd  a  convenient  and  natural  algorithm  for  computing 
the  successive  fractions,  but  in  every  known  instance  the  region  of 
convergence  is  })ractically  the  maximum  for  a  series  of  one  valued 
functions.  The  continued  fraction  of  Ilalphen  [21 , «] ,  so  frequently 
cited  as  an  instance  of  a  continued  fraction  which  diverges  though 
the  corresponding  series  converges,  might  a|)pear  at  first  sight  to 
l)e  an  exception.  But  this  divergence  occurs  only  at  special  i)oints. 
In  fact,  the  continued  fraction  not  only  converges  at  the  center  of  the 
circle  of  convergence  for  the  series,  but,  as  Halphcn  himself  says, 
continues  the  function  over  tiie  entire  })lane  with  the  exception  of 
certain  portions  of  a  line  or  curve.  If  then,  continued  fractions 
offer  such  advantages  for  known  series  and  classes  of  functions,  is 
it  too  much  to  expect  that  in  the  future  tliey  will  throw  a  powerful 
searchlight  upon  the  continuation  of  analytic  functions  and  the 
theory  of  divergent  series? 


*  Les  Series  diveryentes,  }).  GO. 


]54  TJIE  BOSTON  COLLOQUIUM. 

Lkcture  i).     The  Genera/izafion  of  the  CortfinnrtJ  Frarfion. 

In  the  last  lecture  the  algebraic  continued  fraction  was  presented 
under  the  form  of  a  series  of  approximants  for  a  given  function. 
An  immediate  generalization  of  this  conception  can  l)c  obtained 
either  by  increasing  the  number  of  ]ioints  at  which  an  apjn-oxima- 
tion  is  sought  or  by  requiring  a  simultaneous  approximation  to 
several  functions.  The  latter  generalization  results  at  once  from 
an  attempt  to  increase  the  dimensions  of  the  algorithm  or,  in  other 
words,  the  number  of  terms  in  the  linear  relation  of  recurrence 
between  the  successive  convergents  or  approximants.  As  this 
generalization  is  without  doubt  the  more  important,  I  shall  make 
it  the  chief  subject  of  this  lecture.  But  a  few  words,  at  least, 
should  be  devoted  to  the  f  )rmer  extension,  which  is  worthy  of  a 
more  careful  and  systematic  study  than  it  has  received. 

Denote  again  by  A'/.r)  /)(./;)  a  rational  fraction  with  arbitrary 
coefficients.  These  can,  in  general,  be  so  determined  that  its  ex- 
})ansion  at  ,'•  =  0  shall  agree  for  »,  successive  terms  with  a  given 
series 

'•o  +  (*,.'•  +  'V'  +  •  ■  • 

its  expansion  at  .r  —  a^  for  71.,  successive  terms  with 

^1  +  ^^C'"  -  "i)  +  V''"  -  ''1)'  +  •  •  • ' 

at  .'•  =  (I.,  foi-  ii.  successive  terms  with 

/;,  -f-  /•,(.(•  -  a  J  +  lU.r  -  a.,)'  +  •  •  •  , 

and  so  on,  the  total  numbt'r  of  conditions  thus  im|)os(Ml  biMug  equal 
to  y'  +  7  +  1  or  the  number  of  parameters  in  the  rational  frac- 
tion. To  (;ach  s(!t  of  values  foi-  the  /) .  and  7  there  coi-responds  an 
aj)proximaut,  and  the  various  approximants  can  be  ari'anged  into 
a  table  of  multiple  entrv  according  to  tiie  vahies  of  these  ([uan- 
tities.  Continued  fi'actions,  at  least  in  the  ca-^e  ot' a  normal  table, 
can  then  be  obtained  bv  folK)\viug  anv  j)ath  which  j)as.-es  sncces- 
sivelv  from  one  approximant  to  another  contiguous  to  it  but  more 
advanced  in  the  table.  As  we  proceed  along  the  path,  tiie  degi'ee  of 
approxinntion  for  each  of  the  points  O,  "j,  >(„  ■  ■  ■  nni-t  not  decrease 


DIVERGENT  SERIES  AND  CONTINUED  ERACTIONS.     155 

while  at  each  .step  it  is  to  increase  for  at  least  some  one  point.  The 
partial  numerators  of  the  continued  fraction  are  then  either  posi- 
tive integral  powers  of  .k,  x  —  a^,  x  —  a.,,  •  •  •,  or  the  products  of  such 
})0wcrs.  The  degrees  of  the  approximations  obtained  by  stopping 
the  continued  fraction  witli  any  term  can  be  inferred  readily  from 
the  degrees  of  the  partial  numerators  in  .7-,  x  —  a^,  x  —  «,,  •  •  • .  I'he 
details  of  the  theory  have  not  been  worked  out.* 

The  interest  of  such  work  can  perhaps  best  be  made  a})parent 
by  referring  to  the  developments  for  the  simplest  case  in  which 
each  /(.  is  taken  equal  to  1.  The  rational  fraction  ^V,//^,  is  then 
completely  determined  by  the  requirement  that  at  ^>  +  7  -I-  1  given 
points  (t^=  0,  «oj  <(-i,  •••it  shall  take  an  equal  number  of  pre- 
scribed values,  ^Ij,  ^[2,  -I3,  ■  •  • .  If  these  are  the  values  which  a 
single  function  assumes  at  the  points,  we  have  the  rational  frac- 
tions which  were  introduced  by  Cauchy  into  the  theory  of  inter- 
polation [99,  a]  and  which  have  been  quite  recently  formed  into 
a  table  and  examined  by  l\ide  [112].  As  p  +  7  +  1  increases, 
the  number  of  points  at  which  the  approximation  is  sought  like- 
wise steadily  increases. 

When  (J  =  0,  the  rational  fraction  becomes  the  familiar  inter- 
polation-polvnomial  of  JjKjrdngc, 

-^     (^'(^/.)     X  —  a.' 
in  which 

cf)(x)  =  (x  -  a^)  (x  -  a.^  .  .  .  {x  -  ^r^,^,^..,). 

This  has  been  put  into  a  very  interesting  form  by  FroJ>eniii!i  [95] 
which  })ermits,  without  reconstruction, t  of  an  indefinite  increase  in 
the  number  of  its  terms.  Let  us  first  take  1 /(ri  —  .'■)  as  the  par- 
ticular function  of  .f  for  which  an  ai)proximation  is  sought.  From 
the  equations 


*Tlu'  only  investigation  of  tliis  ciiaracter  is  fonnd  in  [7G],  but  on  arconnt  of  the 
nature  of  tlie  functions  there  considered  certain  variations  were  made  in  the  con- 
struction (jf  tlie  table. 

+  Cf.  also  \_W,  '/]. 


156  THE   BOSTON   COLLOQUIUM. 

I      _  1  _       1  x-a^l 

z  -  ,-  ~  [z  -  a^)  -  [x  -  a~)  "  z  -  a^       z  -  a^    z  -  x 

1  X  —  a,  /      1  X  —  (I,,        1 

-f  '  +-  --. 

z  —  ((^  z  —  «i  \ '-  —  ('.1       2  —  (I'o    2;  —  X 
the  sei'ies 

is  immediately  derived,  provided  that  tlie  a.  are  so  distributed  as 
to  fulfill  proper  conditions  for  the  convergence  of  tlie  series.      If 
now  we  take  successively  1,  2,  ?>,  ■••  terms  of  the  expansion,  we 
obtain  the  series  of  polynomials, 

1  ..  1  x-a, 

A.,(.r)  =         -     -  ,       i\^(,,.)  =  -  -f -^  ~-  ,   .  .  . , 

"•  -'       »-(/,'  z-a^       ^z-a;){z-a.y 

and  it  is  evident  that  A'  (x)  for  the  >i  +  1  values  ,'•  =  a,,  <(.,,  •  •  ■,  a  . 
agrees  in   value  with    1  (.:— .r).     By  a])plying    to  (1)  the   well- 
known  formula  oi"  J'Juler  [1,  c/]  *  for  converting  any  intinite  series 
into  a  continuous  fraction  it  follows  immediately  that  these  poly- 
nomials are  the  successive  convergents  of  the  continued  fraction 


1 

x-a^ 

X  —  a.. 

s  -  a, 

z  —  a.. 

z  -  a^ 

1       - 

X  —  (I, 

1  + 

Z  —  (1^ 

X  —  (t., 

-  1  +  .,  _  ^/ 

The  generalization  of  formula  (1)  can   be   made  at  once  in   the 
familiar  manner  by  the  use  of  ('(iiu-lnfs  integral.      We  get  thus 


whicii  i)V  i)lacin(r 


^J-''J  =  (•'•-''■)  (•'■  -".)  •■•(•'•- "J 


may  be  written 

*(•{.   l-:,iri,/:l.,j„i,n,  ,lrr  M„<h.    \V,.<s.,  1  A  :;.  p.  l:;i,  formuhi  (lot). 


DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     157 

For  most  interesting  discussions  of  the  convergence  and  pro])erties 
of  series  having  the  form 

I  may  refer  to  memoirs  by  Frobenius  [95]  and  Boulixson  [99,  c]. 
I  shall  content  myself  here  with  pointing  out  one  simple  appli- 
cation which  is  given  implicitly  by  both  writers  but  has  been 
noted  again  recently  by  Laurent  [103], 

Let /"(.r)  be  any  analytic  fun^ction  the  values  of  w^iich  are  given 
at  a  series  of  points  j^  having  a  regular  point  P  as  their  limit. 
Describe  about  P  as  center  any  circle  C  within  and  upon  which 
f(x)  is  holomorphic,  and  denote  the  points  p.  W'hich  fall  within 
this  circle  by  «,,  a^,  ■  ■  •.  Then  lim  a.  =  P.  If  now  z  describes 
the  perimeter  of  the  circle  and  x  is  a  fixed  interior  point,  the 
series  (1)  will  be  uniformly  convergent  and  consequently  permit 
of  integration  term  by  term.  Equation  (2)  therefore  gives  an 
expression  for  /(.r)  Avhicli  is  valid  in  the  interior  of  C.  This  ex- 
pression shows  at  once  that  an  analytic  function  is  determined 
uniquely  when  its  values  are  known  in  a  sequence  of  points  having 
a  regular  point  P  as  their  limit.  If,  in  particular,  each  /(«,)  =  0, 
f{x)  must  vanish  identically.  In  other  words,  the  zeros  of  an 
analytic  function  can  not  be  infinitely  dense  in  the  vicinity  of  a 
non-singular  })oint.  Further,  BencJixnon  points  out  that  the  con- 
vergence of  the  right  hand  member  of  (2)  is  not  only  the  necessary 
but  the  sufficient  condition  that  f{(f^),  f{n^,  /(^s)?  •  ••  shall  be  the 
values  of  some  analytic  function  at  a  set  of  points  a.  having  a  limit 
point  P. 

AVe  turn  now  to  the  generalization  of  the  algorithm  of  the  con- 
tinued fraction.     The  first  investigation  on  this  subject  is  found  in 


lo8  THE    BOSTON    COLLOQUIUM. 

a  \)a\)er  oi'  Jacobi*  publislicd  ]>osthumniisly  in  1868.  The  devel- 
opments of  Jdcohi  were,  liowever,  of  a  pnrely  numerical  nature. 
On  this  side  tliey  have  been  perfected  recently  by  Fr.  ]\[eycr  [83]. 
Tiie  first  example  of  a  functional  extension  was  given  by  Ilcnnite 
in  his  famous  memoir  [84]  upon  the  transcendence  of  e,  and  the 
theory  has  been  developed  since  independently  of  each  other  by 
J^iiiclierle  and  Pdde. 

To  ex])lain  the  nature  of  the  generalization  it  will  be  desirable 
first  to  refer  to  the  mode  in  which  a  continued  fraction  is  com- 
monly generated.  Two  numbers  or  functions, /^,  ixw({  f^,  are  given, 
from  which  a  secpience  of  other  numl)ers  or  functions  is  obtained 
bv  placing 

{'■'>)  A  =  \A-fv 

A  =  \A  -fv 


in  which  the  X.  are  determined   in   accordance  with  some   stated 
law.      For  the  quotient  ,f\l,f\,  we  obtain  successively 

•^''  -  X          ^    -  X  ^        - 

fl 

and  it  thei'efore  gives  rise  to  the  continued  fraction 

1         1 

\\\  means  ol'  the  e(juati<ms  (.">)  each  /"    ^  can  be  expressed  linearly 
in  tei-m>  of  the  initial  (piantities  Z"^,  /',.      Thus 

in  which  J,,  ,    ,,  .1,  „.,  are  polynomials  in  the  elements  X^..      It  is 
easy  to  >ee  that  these  polviKunials  both  satisfy  the  same  difference 

■  '■  A  llLriiiiciiic  I'hroric  dcr  kcttcnlii'i:rliiilinliclicii  AlL,'(ii'itliiiicn,  in  wclrlioii 
jr,lf  /..ilil  •■111^  (hci  \  uilicr;,'flicnclcii  f^i'liildft  wird.''  .Imini.  pir  Mall,.,  vol.  (V.) 
(  IM',-,  i,   p.   -J'.i. 


I)IVK1{GENT  SERIES  AND  COXTINUED  PR  ACTIONS.     loH 

equation  a«  f.. 

and  for  tlu'ir  initial  values  we  have 

^1,    1=1,  -ln,,=<>, 

Conseciuently  .1,  „  and  —  ^1^,  „  are  the  numerator  and  denominator 
of  the  (//  —  1  )th  eonvergent  of  (4). 

Wiien  the  generating  relations  have  the  form 

./;  =  \.f:  +  /X3./;, 


the  resultant  continued  fraction  is 

^^  '    ^  +  \  +  "" 

A  distinction  then  appears  between  the  system  of  functions 
(.1,  „.^],  —  .1,,,  „_,)  and  the  system  Avhich  consists  of  the  numerator 
and  denominator  of  the  )/th  convergent.  Though  the  quotient 
of  the  two  functions  of  eitlier  system  is  the  ?ith  convergent,  the 
former  jiair  of  functions  satisfy  the  same  relation  of  recurrence  as 
the/',,  namely, 

while  the  corres])onding  relation  for  the  other  system  is 

The  latter  equation  is  called  by  Pincherk  [77,  a]  the  inverse  of 
the  former.  In  the  continued  fraction  (4)  we  took  /U;  =  —  1  so 
that  the  two  relations  were  coincident. 

The  innncdiate  generalization  of  these  considerations  is  obtained 
by  taking  ui  +  1  initial  (piantities  ,fl,J\,  •  •  •,,/"„  iu  place  of  two. 
With  a  very  slight  change  of  notation  we  may  write 


IGO 


THE   BOSTON   COLLOQUIUM. 


Then  by  expressing  J[  in  terms  of  the  m  -f  1  given  quantities  we 
have 

in  which  A:  „  is  a  i)olynomial  in  terms  of  the  X.,  u, ,,  •  ■  •  ,  v., 
('  =  1,   2,  ■  ■  ■ ,  n  —  rn).     These  m  4-  1   polynomials  A^  „  satisfy 
the  same  diiference  equation  (6)  as  the  /^^,  and    for  their  initial 
values  we  plainly  have 

n  =  0  I         0       •  • .      0, 

n  =  l  0          1        • .  •       0, 


n  =  m         0         0       •••       1. 

Hence  they  constitute  a  eom])lete  system  of  independent  integrals 
of  (()).  Furthermore,  in  analogy  with  the  relation  between  two 
successive  convergents  of  (4), 

=   1. 


/>,,  _ 

.,     \-. 

we  have  [8:5,  <i,  \^. 

1  70] 

^lo,„ 

-1,, 

" 

■■      ^^«,» 

(8)            '^y- 

.    ^'h., 

7.1 

••       ^1  „-,,-., 

^■lo,  „. 

.    ^U. 

n  ■  VI 

■  ■   A  „,  ^  „ ..  „j 

=  (_!)- 


The  rehition  which  is  the  inverse  <>f  (*>)  has  the  form 
"^'o  obtain  a  system  of  indcjx'iident   integrals  of  this  ecjuation,  let 


DIVKIKIKXT  SEIIIKS  AND  COXTIXrKD  FRACTIONS.     Kil 

1\,  „  denote  the  minor  of  .1,,.  „  in  (S)^  /^  „  the  minor  of  >1,  „  after 
tiie  first  eohnnn  has  been  mov^ed  over  tlie  remaining  eolnmns  so 
as  to  become  the  last,  J*.,  „  tlie  minor  of  .1^,  „  after  the  tirst  two 
cohunns  iiave  been  moved  over  the  remaining  eolnmns  so  as  to 
))ecome  the  last  two,  and  so  on.  It  can  be  demonstrated  easily 
that  the  desired  system  is  obtained  by  placing  //,„„,  =  i-*,„ 
(/ =  0,  !,  •  • -,  7/i),  and  these  new  polynomials  rather  than  the 
.1,  „  are  the  true  analogues  of  the  numerator  and  denominator  of 
an  ordinary  continued  fraction.  The  connection  between  the  two 
systems  of  polynomials  is,  however,  both  an  intimate  and  a  re- 
ciprocal one,  for  not  only  is  (9)  the  inverse  of  (6)  but  the  converse 
is  also  true.  On  this  account  the  two  systems  can  be  employed 
simultaneously  with  advantage  in  working  with  the  generalized 
continued  fraction. 

For  all  except  the  very  lowest  values  of  ii  the  new  polynomials 
can  be  found  from  the  equations  * 

(9')        1%   +    Kl\„.,  +  f^J\,,^,  +    •  •  •   +  Vj\n-,a  =  J\  .-.-.. 

In  place  of  these  relations  it  will  be  often  found  convenient  to 

employ  such  a  process  as  is  indicated  in  the  following  equations 

for  m  =  2  [83,  a,  p.  180]  .f 

1 

P  P  a  P  '^-'  -  "*"  o 

^  0, 1  -'  0, 2  ,    72, 2  -^  0, ;;  ,  'is,  i 

-'^1,  1  ^    1,  2  72,  1  -^   1,  3  ,      73,2 

'h    1    +    ^ 
73,  1 

1 

72,2    + 

;*  =  7i,,+      -    -",■--'■■■'    ('/...= -^o 

',,.:+       -  ('7,.=  =  -ft) 

,  7i,  1 

'"  •  +  '„.: 

*a.  [83,  «.  p.  174,  eq.  X]. 

t  Cf.   10.  l''iirsteii;in,  "  I'eher    Kt'tteiibriiclu'  luilierer  Onlnuiig";  Jahri'ih,:richt 
uber   d  IS  kuiiii/lichr  Ii'-dliji/uiii'i^iinii  zn    i\'ii'.<')ii  /m  ;    lS7:i  4.      See  also  Scott's   Ik- 
terniiiKnifi',  C'liap.   \',),  |  11-12. 
11 


162 


THE    BOSTON    COLLOQUir.AI. 


AVe  may  thorelorc  very  properly  call  the  system  of  values 


^1 


L 


the  norm  of  a  generalized  continued  fraction,  which  itself  consists 
of  the  computation  of  the  I\  ,^  or  their  ratios. 

To  apply  this  generalization  to  the  construction  of  algebraic 
continued  fractions,  it  is  only  necessary  to  select  as  the  m  +  1  initial 
functions  /„  •  •  • ,  J\^  series  in  ascending  powers  or  series  in 
descending  powers  of  x.  The  nature  of  the  ensuing  theory  will 
be  explained  suliiciently  by  develoi)ing  here  the  simplest  case,  in 
which  three  such  series  are  given  [77,  c]      Take  then 


^  X  X'  .1" 


s. 


X  X 


If  we  next  place 

the  coetlieients  rr^,  a[^,  h^^  can  be  so  determined  that  S^  shall  begin 
with  at  least  as  high  a  power  of  1/./:  as  the  third.  Xormally  the 
degree  is  exactly  3,  and  similarly  lor  each  consecutive  value  of  n 
we  have 

in  which  .S'  denotes  a  series  beginning  with  the  /)th  power  of 
l/x.  Hence  unless  certain  sju'cilied  conditions  are  satistied,  a 
regular  continued  fraction  will  be  obtained  having  the  norm  : 

!     1        d^j'  -f  d'^^       A^i 


A, 


(; 

A^^ 

A 

7?„ 

C[_^ 

..K.. 

.1 

-JK-r. 

DIVERGENT  SERIES  AND  CONTINUED  FRACTIONS.     163 

This  iKirm  will  not  be  altered  in  any  way  by  dividing  (10) 
throiitrh  bv  H  .  It  is  tlierefore  determined  nni(iiielv  bv  the  ratios 
of  S^,  >S',,  ^S".,,  and  conversely  the  ratios  by  the  norm. 

Without  loss  of  generality  we  may  set  S^  =  1.      Place  also 

/>'      c 

If  then  H  +  o  in  (11),  is  replaced  successively  by  n  and  n  -\-  1, 
and  the  two  equations  are  solved  for  S^  and  ^S'^,  we  obtain 

or 

(12)  '^^  -  ?; = F      (^-^^  ^^'^  ^"^'^  -  ^'  ^^'-^^)' 

and 

(13)  '5.-t;=M"  (fM  =B  S    ,-B  .,S). 

An  examination  of  i""^^,  Q^^,  R,  X  ,  ^^^  will  show  that  their  degrees 
in  X  are 

71  —  1 ,  ??  —  2,  /)  —  3,   —/•—],   —  /•.  (n  =  2r), 

n  —  \,   n  —  '2,  ?(  — 3,    — /•— i,    — /•— 1  (»  =  2/-+l). 

Hence  the  expansions  of  Q^  P^^  and  R  P^^  in  descending  powers  of 
X,  agree  with  >S'j  and  »S'2  to  terms  of  degree  3/'  —  1  and  3/-  —  2  in- 
clusive if  /i=2r,  and  of  the  3/'th  degree  if  ?i  =  2r+l.  The 
generalized  continued  fraction  therefore  aifords  a  solution  of  the 
problem  :  to  find  two  rational  fractions  with  a  common  denom- 
inator which  shall  give  as  close  an  approximation  to  the  given 
functions  N,  and  *S'.,  as  is  consistent  with  the  degrees  prescribed  for 
their  numerators  and  denominators. 

When  three  series  in  ascending  powers  of  x, 

s,  =  k-  +  k;>x  +  k:\r  +  ...  (/=  1,2,3), 


164  THE   BOSTON   COLLOQUIUM. 

are  choseu  as  the  initial  functions,  a  more  comprehensive  algorithm 
can  be  introduced.  Fade  [79,  a]  takes  three  polynomials  Af/{x), 
^ ■;?(«),  yl-f„'(a;)  with  undetermined  coefficients,  the  degrees  of  which 
are  indicated  by  their  subscripts,  and  requires  that  their  coefficients 
shall  be  so  determined  that  the  expansion  of 

in  ascending  powers  of  x  shall  begin  with  as  high  a  power  as 
possible.  Ordinarily  this  is  the  [p  +  p'  +  p"  +  2)th  power.  To 
each  set  of  values  of  p,  p' ,  pj"  he  shows  that  there  corresponds 
uniquely  a  group  of  three  polynomials  which  he  terms  the  "  asso- 
ciated polynomials,"  and  these  groups  he  arranges  into  a  table  of 
triple  entry  according  to  the  values  of  p,  p' ,  pj".  An  exactly 
similar  table  can  not  be  constructed  for  three  series  in  descend- 
ing powers  of  x,  inasmuch  as  the  substitution  of  l/x  for  x  in 
A^l\  ■  ■  ■ ,  A'V,  gives  three  rational  fractions,  with  powers  of  .?■  in 
the  denominators  which  can  not  be  thrown  away  unless 

p=p'=.p". 

The  new  table  is  handled  l)y  l^ade  in  the  same  manner  as  the 
one  ])reviously  constructed  for  a  single  series.  In  particular,  he 
examines  the  relations 

a.l;; '  +  ^A\;>  +  jA;^  =  Ap  (/  =  1 ,  2,  3), 

which  exist  between  four  successive  groups  of  associated  poly- 
nomials, a,  /3,  7  being  rational  functions  of  x  which  are  indepen- 
dent of  the  value  of  /.       ^^'hen  it  is  possil)le  to  so  select  a  se(juence 

•  •  ■  A^y,  A",  A'',  Ay,  Ay,  ■  ■  ■  that  a,  /3,  7,  are  ])olynomials  of 
invariable  degree  for  any  four  consecutive  terms  in  the  sequence, 
the  scijucncc  or  continued  fraction  is  said  to  be  regular.  In  a 
normal  tabh'  there  are  found  to  be  four  distinct  types  of  such  con- 
tinued fi-actions.  It  is  worth  noting,  howevei-,  that  the  diagonal 
ty|)o  which  was  the  best  in  an  ordinarv  table,  no  longer  exists  since 
it  is  found  that  when  the  seijUenee  fills  :i  diagonal  file  of  the  tal)h', 
o(,  f-^,  and  7  are  110  longer  polynomials  but  rational  fractions  having 
a  eonuiioii  denominator. 


DIVERGENT  SERIES  AND  CONTINUED  ERACTIONS.     165 

In  one  important  respect  I\hWh  investigation  has  a  narrower 
reach  than  Plncherle\s  and  needs  completion.  The  existence  of  a 
second  group  of  associated  polynomials  —  the  P,  Q  ,  R^^  of  7V/i- 
cherle  —  is  not  brought  to  light.  As  has  been  already  pointed  out, 
it  is  this  second  grouj)  of  polynomials  which  is  the  true  analogue 
of  the  convergent  of  an  ordinary  continued  fraction  and  which 
must  take  precedence  in  considering  the  convergence  of  the  algo- 
rithm or  the  closeness  of  the  approximation  afforded  to  the  initial 
functions.  Pincherle's  definition  of  convergence  [82]  is  not,  how- 
ever, so  framed  as  to  require  explicitly  the  introduction  of  these 
polynomials.     If  the  given  system  of  difference  equations  is 

(1^)  /„  +  3  =  ^„/,-  +  2  -f  <./;,4-.  +  /«  (^^  =  0,  1,  2,  . . .), 

the  continued  fraction  is  said  by  him  to  be  convergent  when  the 
two  following  conditions  are  fulfilled  : 

(1)  There  is  a  system  of  integrals  F,  F',  F'^  of  (14)  such  that 
I\/J']^,  F'll F ^^  have  limits  for  n  =  oo,  and  these  limits  are  different 
from  0. 

(2)  There  is  also  one  particular  integral — called  by  Pincherle 
the  integrak  disihito  —  the  ratio  of  which  to  every  other  integral 
of  (14)  has  the  limit  zero. 

Pincherk^'i  interest  is  evidently  concentrated  upon  this  prin- 
cipal integral.  It  seems  to  me,  however,  more  natural  to  call 
the  algorithm  convergent  when  the  ratios  QJP„  and  P,JP,^  (cf. 
Equations  12  and  13)  converge  to  finite  limits  for  n=  oo.  Under 
ordinary  circumstances  these  limits  will  doubtless  coincide  with 
the  ratios  of  the  generating  functions,  ^fjj^  and  f.,l_f^^. 

In  the  case  of  an  ordinary  continued  fraction  the  two  definitions 
coalesce.  For  suppose  that  the  7;th  convergent  X^JJ)^^  of  (4')  has 
the  limit  L.  Then  .V  —  LI)  is  such  an  integral  of  the  differ- 
ence equation, 

./;,  =  \,./;,-,  +  f^j„.„ 

that  its  ratio  to  any  other  integral,  L\N^  -f  l:.J\,,  has  the  limit  0. 
Conversely,  if  the  ])rincipal  integral  A  —  Lf\^  exists,  there  must 
be   a   limit   L  for  the  continued   fraction.      Possiblv  the  case  in 


166  THK    BOSTON    COLLOQUIUM. 

\vliieh  the  ])riiici])al  integral  is  D^^  might  be  called  an  excep- 
tion, since  the  continued  fraction  is  then  convergent  by  PinelLerk's 
definition,  but  lim  ^V,/^„  =  oo. 

A  study  of  the  conditions  of  convergence,  so  far  as  I  am  aware, 
has  at  ]')resent  l)een  made  in  only  two  special  cases.  Fr.  Meyer 
[83,  a,  §  7]  has  made  a  partial  investigation  when  the  coefficients 
X^,  •  •  • ,  v^^  in  equations  (<J)  are  negative  constants.  PincJterk  [82] 
has  examined  the  case  in  which  the  coefficients  of  the  recurrent 
relation 

./:  +  (^'.»'  +  <)./;,-Hi  +  ^,./:,.i-2  =/,.-. 

have  limiting  values  and  finds  that  the  generalized  continued  frac- 
tion is  convergent  for  sufiiciently  large  values  of  x.  Let  the  limits 
of  the  coefficients  be  denoted  by  a,  a',  and  b  respectively.  To 
demonstrate  the  convergence  he  avails  himself  of  the  notable  the- 
orem of  Poincare,  already  cited  in  I^ecture  4.  If,  namely,  no  two 
roots  of  the  equation 

(15)  z^  -  bz'  -  (ax  +  a')f~  1  =  0 

are  of  equal  modulus, _/^//]^_j  will  have  a  limit  for  /(  =  oo,  and  this 
limit  will  be  one  of  the  roots  of  the  auxiliary  equation  (15), 
usually  the  root  of  greatest  modulus.  From  this  it  follows  di- 
rectly that  AJA^^_^,  BJ  Ji^^_^,  ^',/^l-i  iis  (piotients  of  integrals  of 
the  diiference  equation  last  given,  also  .PJJ\_^,  Q,,J  Q.,^^\,  PJP,,^] 
as  integrals  of  the  inverse  ecjuation,  have  each  a  definite  limit.  The 
existence  of  limits  for  QJI\^  and  of  U^J  I\^  is  then  esta})lished 
for  sulficiently  great  values  of  x,  and  the  analytic  cliaracter  of 
these  limits  is  fhially  argued.  Let  them  be  denoted  by  l\y)  and 
r(u-).  Then  X^=  A^  +  P„r{->')  +  ('JV)  is  the  principal  in- 
tegral of  the  difference  e(juation,  and  has  the  following  distinctive 
proj)ertv  :  Its  expansion  in  |)()wers  of  1  jx  begins  with  the  highest 
])()ssil)le  power  consistent  with  tlie  degrees  of  vl^,  Jl^,  T  ,  and 
ciiiiicides  with  /"   for  each  successive  value  of /^ 


DIVKKGEXT  SERIES  AND  (  ( )N  PI  N  ri:i)  FRACTIONS.     KIT 
BllUJOOUAPIIV    OF    MkMOIRS    UKLAriN(i    TO    AuJHHRAIc 

(  'on  TIM" i: I)    Fkaotions. 

In  the  following  l)il)lioi2:ra])hy  only  works  in  I^atin,  Italian, 
French,  (iernian,  and  Knolish  are  inclnded.  In  I(  o/////^r/',s  J/r?/Ac- 
mat'iKvJu'r  Hi'ichcrsclKftz  (heading  KcttenbrWclie^  several  dissertations, 
etc.,  are  mentioned  which  may  possil)ly  relate  to  algebraic  con- 
tinned  fractions  bnt  which  are  not  accessible  to  tlie  writer.  They 
are  therefore  not  inclnded  here.  The  writer  wonld  be  glad  to 
have  his  attention  called  to  any  noteworthy  omissions  in  the 
bibliography. 

In  many  cases  it  has  been  extremely  difficnlt  to  draw  the  line 
between  inclusion  and  exclusion,  especially  under  divisions  vi-ix. 

Any  classification  of  the  material  which  may  be  adopted  will  be 
open  to  objections,  but  even  an  imperfect  classification  will  prob- 
ably add  greatly  to  the  usefulness  of  the  bibliography.  Since 
much  of  the  work  relating  to  algebraic  continued  fractions  ap])ears 
elsewhere  under  other  headings,  it  is  believed  that  such  a  bibliog- 
ra]>hy  as  is  here  given  may  be  of  service. 

For  a  brief  resume  of  the  theory  of  algebraic  continued  frac- 
tions the  reader  is  referred  to  Os(/oo(rs  section  of  the  Eiicyklopad'ie 
der  yrath.   Wisscnschaff,  ii  b  i,  §§  38-30. 

I.     On  thk  Derivation  of  Continted  Fractions  from  Power 
Series.     General  Theory. 

A.    Early   Wo7'ks. 

1.  Euler.     (a)  Iiitroductio  in  analysin  infiiiitorum,  vol.   1,  chap.  18, 

1748. 
(b)  Do   transformatione  serieruin  in  fi-actiones  continuas.     Opus- 
eulu  analytica,  vol.  2,  pp.  138-177,  1785. 

2.  Lambert,     (a)  Yerwandluiig  der  Briiche.     Beytriige  zum  Gebrauche 

der  ^Mathematik  und  deren  Anweudung,  vol.  2,,  p.  54  ft'.,  p.  161, 
1770. 
(b)  ]\Iemoire  sur  quelqucs  proprietes  remarqnablcs  des  quantit^s 
transceudentes    circulaires    et     logarithmiques.       Histoire    de 
TAcad.  roy.  des  sciences  et  belles-lettres  k  Berlin,  1768. 


16<S  TIIK    BOSTON    COLT.OQriUM. 

8.  Trembley.      Eechcrclu'S    sur    les    iractioiis    continues.      3Iem.    de 
TAcud.  roy.  de  Berlin,  1794,  pp.  109-142. 

4.  Kausler.     («)  Expositio  methodi  series  cpiascuiKpie  datas  in  fi'ac- 

tiones  continuas  convertendi.      Mem.   de  I'Aead.  imp.  des  sci- 
ences de  St.  I'etersbourg,  vol.  1,  pp.  156-174,  1802. 
(b)  De  insigni  usu  fractioimm  coiitinuanim  in  calculo  integrale. 
Ibid.,  vol.  1,   pp.  181-194,  1803. 

5.  Viscovatov.     (a)  De  la  methode  generale  pour  reduire  toutes  sortes 

des  ((uantites  en  fractions  continues.     Ibid.,  vol.  1,  pp.  226-247, 
1805. 
(b)  Elssai  (Vune  metbode  generale  pour  reduire  toutes  sortes  de 
series   eu   fractions  continues.     Nova  Acta  Acad.  Scient.  imp. 
Petropolitanic,  vol.  15,  pp.  181-191,  1802. 

6.  Bret.      Theorie    generale    des    fractions    continues.      GJergonne's 

Annales  de  3Iatb.,  vol.  9,  pp.  45-49,  1818.     Unimportant. 

7.  Scubert.      De    transformatione    seriei    in    fractionem    continuam. 

Mem.  de  TAcad.  imp.  des  sciences  de  St.  Petersbourg,  vol.  7, 
pp.  139-158,  1820. 

8.  Stern,     (o)  Zur  Tbeorie  der  Kettenbriiche  und  ihre  Anwendung. 

.Jour,  fur  Math.,  vol.  10,  pp.  241-265,  1833. 
(6)  Zur  Theorie  der  Kettenbriiche.     Jour,  fiir  Math.,  vol.  18,  pp. 
G9-74,  1838. 

9.  Heilermann.     (a)  Ueber  die  Yerwandlung  der  Reilien   in   Ketten- 

briiche.    Jour,  fiir  Math.,  vol.  33,  i)p.  174-188,  1846  ;  also  vol. 
46,  pp.  88-95,  1853. 
(b)  Zusammenhang  unter  den  Coetlicienten  zweier  gleichen  Ket- 
tenbriiche von  ver.scliiedener  Form.      Zeitschrift  fiir  INIatli.  und 
I'll  vs..  vol.  5,  i)p.  362-363,  1860.      Unimportant. 

10.  Hankel.     l'el)er  die  Transformation  von  lleihen  in  Kettenbriiche. 

Berichte  der  Sachischen  (Jesellschaft  der  Wissenschaft  zii  Lei])- 
zig,  vol.  14,  1)]).  17-22,  1862. 

11.  Muir.      ((()  On  tlie  transformation  of  ddii.^s'  liypergeometric  series 

into  a  continued   fraction.      Proc.   of  the   London   ]Math.   Soc, 
vol.  7,  ])p.  112-118,  1876. 
(h)   New  general  formula'  for  tlie  transfbrmation  of  infinite  series 
into  continue*!  IVaclions.      Trans,  of  the  11.  Soc.  of  Ildinbiirgh, 
vol.  27,  i\K   167-471.  1876. 

The  general  fuiiniila'  in  these  menioiis,  which  3Iuir  snjjjjosed 
to  he  new,  iiad  ixen  pre\iously  gi\'en  by  Heilerniann  in  'J{(i). 

12.  Heine.      Ilandbiich  dei'    l-Cugelfunction,  2''  .\uflage,   1878;   cliap.  5, 

JMc   Kcttciibi  iiche,    pp.  260    2'.t7. 

Thi>  gi\(s  a  good  idea  of  the  state  of  tlie  theory  ii]»  to  1N78. 


DIVKKGENT  SERIES  AND  CONTTXrEI)  FRArTIONS.     1G!» 

B.   The  Modern  Tlieonj. 
The  beginniugs  of  tliis  theory  are  to  be  fouiul  iu  Xos,    110 
and  HI. 

13.  Frobenius.     I'eber  Relationeu  zwisclien  den  Naherungsbriiclien  von 

Potenzreilien.     Jonr.  fiirlNIatli.,  vol.  90,  pp.  1-17,  1881. 

This  fundamental  memoir  marks  an  im])ortant  advance.     See 

1G(<0. 

14.  Stieltjes.     Sur  la  reduction  en  fraction  continue  d'une  serie  pro- 

cedant  suivant  les  puissances  descendantes  d'une  variable. 
Ann.  de  Toulouse,  vol  3,  H,  pp.  1-17,  1889. 

15.  Pincherle.     Sur  une  application  de  la  theorie  des  fractions  contin- 

ues algebriques.     Comp.  Rend.,  vol.  108,  p.  888,  1889. 

16.  Fade,     (a)  Sur  la  representation  approchee  d'une  fouction  par  des 

fractious  ration nelles.  Thesis,  published  in  the  Ann.  de  rp]c. 
Nor.,  ser,  3,  vol.  9,  supplement,  pp.  1-93,  1892. 

This  very  fundamental  memoir  is  the  best  one  to  read  for  the 
purpose  of  learning  the  elements  of  the  theory  of  algebraic 
continued  fractions.  The  same  point  of  view  is  taken  as  by 
Frobenius  in  (13)  and  is  more  completely  developed.  The 
thesis  was  preceded  by  the  two  following  preliminary  notes  : 

(«')  Sur  la  representation  approchee  d'une  fonction  par  des 
fractions  rationnelles.     Comp.  Rend,  vol.  Ill,  p.  674,  1890. 

(a'^)  Sur  les  fractions  continues  regulieres  relatives  a  e''. 
Comp.  Rend,  vol.  112,  p.  712,  1891. 

(h)  Recherches  nouvelles  sur  la  distribution  des  fractions 
rationnelles  approchees  d'une  fonction.  Ann.  de  I'Ec.  Nor., 
ser.  3,  vol.  19.  pj).  153-189,  1902. 

(c)  Aperyu  sur  les  develox>pements  recents  de  la  theorie  des 
fractions  continues.  Compte  rendu  du  deuxieme  Cougres  inter- 
national des  mathematiciens  tenu  a  Paris,  pp.  257-264,  1900. 

Only  a  resti'icted  portion  of  the  field  is  here  reviewed,  and  in 
this  ])ortion  the  important  work  of  Pincherle  is  overlooked. 

17.  Fade,     (a)  Sur  les  series  entieres  convergentes  on  divergeutes  et 

les  fractions  continues  rationelles.  Acta  ^latli.,  vol.  18,  pp. 
97-111,  1894. 

(a^)  Sur  la  possibilite  de  definir  une  fonction  par  une  serie 
entiere  divergente.      Comp.  Rend.,  vol.  110,  p.  686,  1893. 
"lee  ali^o  No.  2i>a,  76. 

II.       Ox    CONVEKGKXCK. 

(For  a  resume  of  the  criteria  for  the  convergence  of  continued 
fractions  with  real  elements  see  1'kiX(:shkim"s  rejtort  in  ihc  En- 
cyklopadie  der  mathematischcn  Wissenscliaften,  I  A  3,  p.  126,  ff.) 


ITO  THE    BOSTON    COT .LOQITUM. 

IS.  Riemann.  Sullo  svolgimeuto  del  (|uozicnte  ili  due  serie  ipergeo- 
nietrielie  in  frazione  continua  iiifinita,  1868.  (resammelte  math- 
einatisehe  Werke.  ])p.  400-400. 

18.  hi.'i.  Worpitzky.     Vntersuehung  liber  die  Ent\vickelun<i-  der  mono- 

droinen  und  monogenen  Fimctionen  durch  Kettenbriielie.     Pro- 
grainm,  Friedrichs  Gymnasium  und  Realschnle,  Berlin,  1865. 

This  program  and  the  two  following  memoirs  of  Thome  were 
published  before  Riemann' s  posthumous  fragment. 

19.  Thome,     (a)  Ueber  die  Kettenbruchentwickelung  der  Gauss'schen 

Function  F{o.,  1,  },  .r).     Jour,  fiir  Math.,  vol.  66,  pp.  322-336, 
1866. 
(h)  Ueber  die  Kettenbruchentwickelung  des    Gauss'schen   Quo- 

tienten 

F(a,3J,l,r+l,.r) 

H",  ^  >,  ^■)  ' 

Ibid.,  vol.  67,  pp.  299-309,  1867. 

20.  Laguerre.     Bur  I'integrale 


.r 


dr. 


Bull,   de  la  Soc.   :\Iath.   de  France,  vol.  7,  pp.  72-81,  1879,  or 
Oeuvrcs,  vol.  1,  p.  428. 

Historically  an  important  memoir  because  of  its  develojjment 
of  the  connection  between  a  divergent  power  series  and  con- 
vergent continued  fraction.  See  the  first  footnote  in  lecture  4  ; 
also  No.  102,  p.  30. 

21.  Halphen.     (a)    Sur   la  convergence  d'une  fraction   continxie    alg^- 

brique.     Comp.  Rend.,   vol.  100  (1885),  p]).  1451-1454. 

(ft)  Same  subject.     Ibid.,  vol.  106  (1888),  pp.  1326-1329. 

(c)  Traite  des  foncti(Mis  ellipti(iues.      Chap.    14.     Fractions    con- 
tinues et  integrales  i)seudo-elli])ti(|ues. 

22.  Pincherle.     Alcuni  teoremi  sulle  fra/.ioni  continue.      Atti  delle  R. 

Accad.  dei   Lincei,  ser.  4,  vol.  5,,  ])]>.  640-643,  1889. 

The  test  for  convergence  given  here  is  included  in  a  moro 
gencial  criterion  given  later  by  I'ringslieim,  No.  29. 

23.  Pincherle.      Sur  les  fractions  continues  algebricjues.      Ann.  de  I'Ec. 

Nor.,  ser.  3.  vol.  6.  ]>]).  145-152,  1889. 

All  iii('oiii])iete  result  is  here  ol)tained.  See  No.  32c  for  the 
coiii]il('t('  theorem. 

24.  Fade.      Sur  hi  convergence  d<'s  fractions  continues  simjdes.      Comp. 

Rend.,  vol.  112,  p.  988,  1891.     Also  found  in  §5^45-47  of  No.  16a. 


DIVEKCrENT  SERIES  AND  CONTINUED  FPvACTTONS.     171 

25.  Banning.     Uebor  Kutrol-  und  ("yliinlorfuiiktioneii   iin<I  deren  Ket- 

teiibruchentwickclung.     Dis.sertatlon,  Bonn.  1S04.  pp.  1-38. 

26.  Stieltjes.     (a)  Recherelies  sur  les  fractions  continues.      Annales  de 

Toulouse,  vol.  8,  J.  p]).  1-122.  and  vol.  9,  A.  j*]).  1-47.  1894-95. 
Published  also  in  vol.  32  of  the  ^Nlenioires  presentes  a  I'Acad. 
des  sciences  de  I'Institut  National  de  France. 

A  rich  memoir,  develo])ing  particularly  the  connection 
between  an  important  class  of  continued  fractions  and  the  cor- 
responding integrals. 

(a')  Sur  \\n  developpement  en  fraction  continue.  Comp. 
Rend.,  vol.  99,  p.  508,  1884. 

(a'O  Same  subject.     Ibid.,  vol.  108  (1889),  p.  1297. 

(rt''^)  Sur  une  application  des  fractions  continues.  Ibid.,  vol. 
118  (1894),  p.  1315. 

(a")  Recherches  sur  les  fractions  continues.      Ibid.,  vol.  118 
(1894),  p.  1401. 
Markoff.     (b)  Note  sur  les  fractions  continues.     Bull,  de  I'Acad. 
imp.   des    sciences  de  St.  Petei'sbourg,  ser.  5,  vol.  2,  pp.  9-13, 
1895. 

This  gives  a  discussion  of  the  relation  of  his  work  to  that  of 
Stieltjes. 

27.  H.  von  Koch,     (a)  Sur  un  theoreme  de  Stieltjes  et  sur  les  fonctions 

definies  par  des  fractions  continues.     Bull,  de  la  Soc.  Math,  de 
France,  vol.  23,  pp.  33-40,  1895. 

(a^)  Sur  la  convergence  des  determinants  d'ordre  infini  et  des 
fractions  continues.     Comp.  Rend.,  vol.  120,  p.  144,  1895. 

28.  Markoff.     Deux  demonstrations  de  la  convergence  de  certaines  frac- 

tions continues.     Acta  Math.,  vol.  19,  pp.  93-104,  1895. 

Contained  also  in  his  Differenzenrechnung  (deutsche  Ueber- 
setzung),  chap.  7,  §  21-22. 

This  discusses  the  convergence  of  the  usual  continued  frac- 
tion for 

fi!/)d!/ 
If 


£■[ 


when/(!/)  >  0  between  the  limits  of  integration. 
29.  Pringsheim.     I'eber    die    Convergenz    unendlicher    Kettenbriiche. 
Sitzungsberichte   der   math.-phys.    Classe    der  k.    bayer'schen 
Akad.  der  Wisseuschaften,  vol.  28,  pp.  295-324,  1898. 

The  most  comprehensive  criteria  for  convergence  yet  obtained 
are  found  in  29,  31,  and  326. 


172  TH1-:  BOSTON'  c"OLT.OQrrr:M. 

30.  Bortolotti.     Siilla  0')nveri!,enz;i  delle  fVazioui  contiime  alfjehriche. 

Atti  (lella  K.  Accad.  dei  l.inci-i.  sor.  5.  vol.  8,,  ])p.  28-38,  1899. 

31.  Van  Vleck.     On  the  convcrgeiu'e  of  continued  IVactionf^  with  cora- 

])lex  elements.     Trans.  Anier.  ^Nlath.  Soc,  vol.  2,  ])p.  215-233, 
1901. 

32.  Van  Vleck.     (a)  On   the  converjienee  of  the  eoiitinucd  fraction  of 

(iauss  and  other  continued  fractions.     Annals  of  Math.,  ser.  2, 
vol.  3.  pp.  1-18,  1901. 
(h)  On  the  convergence  and  character  of  the  continued  fraction 

a^z       a.,z       a.^z 
1  ^    1    -i-  1  -      *  '  ' 

Trans.  Amer.  Math.  Soc,  vol.  2,  pp.  476-183,  1901. 
((')  On  the  convergence  of  algebraic  continued  fractions  whose 
coefficients  have  limiting  values.      Ibid.,   vol.   5,  pp.   253-262, 
1904. 

33.  Montessus.     («)  Sur  les  fractions  continues  algebriques.      Bull,  de 

la  Soc.  Math,  dc  France,  vol.  30.  pp.  28-36,  1902. 
The  content  of  this  memoir  was  discussed  in  lecture  5. 
(h)  Same  title.     Oomp.  Eend.,  vol.  134  (1902),  p.  1489. 
See  al-'«)  '->7(i\  41. 

III.    On  Various  Contixukd  Fractions  of  Special  Form. 

A.    The  Continued  Fraction  of  Oanss. 

34.  Gauss.     Dis([uisitiones  generales  circa  seriem  infinitam 

a.  3  „(.  .^  1).9(.3  a.  1) 

1>-  l-2}(v^-l) 

Deutsche  Uebersetzung  von   Simon,  or  Werkc,  vol.  3,  jtp.  134- 
i:'.s,  1S12. 
34.  hi-'<.   Vorsselman  de  Herr.      Sjiecimen  inaugurale  de  fi'actionibus  con- 
tinni-.      Dissertation,   rtrcchl.  1S33. 

Numerous   rercrcnccs   are  given   lierc   to   the  early  literature 
u|Min  contiiiunl  fractions. 
31.  /'■/•.   Heine.      Auszug  eines  Sclireilx-ns  iiber  Kettenbriiche  von  Herrn 
1'].   Heine  an   den    IIerausgel)cr.      .lour,  fiir   Math.,  vol.  53,   ])]). 
•Jsl    L's.-),   1,S57. 

Sre  <l].«,  40r.    p.    L':;l. 

;',5.  Euler.  (n)  ( 'oiniiinitat  io  in  iVact  ioncni  continunni  in  (|Ua  illustris 
fjujrdni/e  p(jt fstate.--  binoinialrs  (•.\])i-cs.sit.  Memoires  de  I'Acad. 
imp.  (les  sciences  de  St.   reter-lioui-g,  vol.  6,  jiji.  3-11.  ISIS. 


DIVERGENT  SERIES  AND  CONTINrED  FRACTIONS.     173 

Pade.  (b)  Sur  la  gc'nerali.siition  des  devcloppements  en  fractious 
continues,  donnes  par  Gaus!<  et  par  lufler,  de  la  function 
(1  --  x)'".     Conip.  Rend.,  vol.  129,  p.  753,  1899. 

(c)  Sur  la  generalisation  des  developpements  en  fractions  contin- 
ues, donnes  par  Lagrange  de  la  fonction  (1  -f-  .r)'".  Ibid.,  vol. 
129,  p.  87-5,  1899. 

(cZ)  Sur  I'expression  generale  de  la  fraction  rationuelle  approchee 
de  (1  -^  x)'".     Ibid.,  vol.  132,  p.  754,  1901. 
See  also  Xos.  11,  32(i,  65. 

B.    The  Co/itiiiued  Fractions  for  e^ 

36.  Winckler.     Ucber  angeniilierte  Bestinimungen.     Wiener  Berichte, 

Math.-naturw,  Classe,  vol.  72,  pp.  646-652,  1875. 

37.  Pade.     (a)  Memoire  sur  les  developpements  en  fractions  continues 

de  la  fonction  exponentielle,  pouvant  servir  d'introduction  a  la 
theorie   des   fractions   continues   algebriques.      Ann.    de    I'Ec. 
Nor.,  Ser.  3,  vol.  16,  pp.  395-426,  1899. 
(a^)  Sur  la  convergence  des  reduites  de  la  fonction  exponentielle. 
Coinp.  Rend.,  vol.  127,  p.  444,  1898. 
See  also  Xos.  16a".  106,  and  pages  243-5  of  40c. 

C.  The  Continued  Fraction  of  Bessel. 

38.  Giinther.     Bemerkungen    iiber   Cylinder-Functionen.     Archiv   der 

Math,  und  Piiys.,  vol.  56,  pp.  292-297,  1874. 

39.  Graf,     (a)  Relations  entre  la  fonction  Besselienne  de  1""  espece  et 

une  fraction  continue.    Anuali  di  Mat.,  ser.  2,  vol.  23,  ])p.  45-65, 
1895. 

Giving  references  to  earlier  works  where  the  continued  frac- 
tion of  Bessel  is  found. 
Crelier.     (b)  Sur    quelques   proprietes    des  functions    Besseliennes, 
tirees  de  la  tlieorie  des  fractions  continues.     Annali  di  Mat., 
vol.  24,  pp.  181-163,  1896. 
See  also  Nos.  25,  32a. 

D.  TJie  Continued  Fraction  of  Heine. 

40.  Heine,     (a)   Ueber  die  Reihe 

^  {<t  - 1)  iq^  - 1)    ^  (r  - 1)  (r  - '  - 1)  (<i^  ~  1)  i<i^  ■  •  - 1)  ., 
(^/-i)('y>-i)"''"      ('^-i)('/-'~i)(7^-i)('y^ -'-i)    ''^"" 


174  THE   BOSTOX   COLLOQUIUM. 

Jour,  fur  Math.,  vol.  32,  pp.  210-212,  1846. 

(b)  Uutersuchuug  u])er  die  (selbe)  Reihe.    Ibid.,  voL  34,  pp.  285- 
328,  1847. 

(c)  Ueber  die  Ziihler  und  Nenner  der  Nilherungswerthe  von  Ket- 
tenbriiche.     Ibid.,  vol.  57,  pp.  231-247,  1860. 

Christoffel  [d)  Zur    Al)haudlung    "Ueber    Ziihler    und    Nenner" 
(u.  s.  w.)  des  vorigen  Bandes.     Ibid.,  vol.  58,  i)p.  1)0-91,  1861. 

41.  Thomae.     Beitnige  zur  Theorie  der  durch  die  Heiue'sche   Reihe 

darstellbaren  Funktioneu.    Jour,  fiir  Math.,  vol.  70,  1869.    See 
pp.  278-281  where  the  convergence  of  Heine's  continued  frac- 
tion is  proved. 
See  also  32a. 

42.  (On  Eisensteiii's  continued  fractions). 

Heine,     (a)  Verwandlung  von  Reihen  in  Kettenbriiche.     Jour,  fiir 
Math.,  vol.  32,  pp.  205-209,  1846. 
See  also  vol.  34,  p.  296. 
Muir.     (6)  On  P^isenstein's  continued  fractions.     Trans.  RoJ^  Soc. 
of  Edinburgh,  vol.  28,  part  1,  pp.  135-143,  1877. 

Muir  plainly  was  not  aware   of  the   preceding   memoir    by 
Heine. 

E.    The  Continued  Fraction  of  Stieltjes.     {See  No.  26.) 

43.  Borel.     Les  series  de   Stieltje.f,    Chap.   5  of  his  Memoire  sur    les 

series divergentes.  Ann.  del'Ec.  Nor.,  ser.  3,  vol.  16,  pp.  107- 
128  ;  and  also  chap.  2  of  his  treatise,  Les  Series  divergentes, 
pp.  55-86,  1901. 

44.  Pade.    Sur  la  fraction  continue  de 'SV;>/(yV'.s-.    Comj).  Rend.,  vol.  132, 

p.  911,  1901. 

45.  Van  Vleck.       On    an    extension    of  the    1894    memoir    of  Stieltjes. 

Trans.  Amer.  Math.  Soc.  vol.  4,  pj).  297-332,  1903. 
See  al.^0  Nos.  27,  102. 

F.    The  Continued   Fraction  for 

1    ■-  mr   i'  //(("'    •-  ")•''"   i'  "K"'    '■'  ")  ("'    ■    2/!).r'  -|-  •  •  • 

and  its  sjterial  (v^^r•-^■. 

46.  Euler.     (a)  \)v  sfriel)us  (iivci-gentil)us.      Xovi  commentarii  Acad. 

scicnt iarum  iinjicrialis  rflr<)})()]itun;e,  vol.  5,  j)p.  205-237,  1754- 
5  :  ill  particiilai'  pp.  225  and  2.">2-2l'>7. 
(//)    l)c   t ransfui'matioiif  serifi  divergcntis 

1  —»/(.(•    ■    ii)(iii    •    n).r"  ~-  ni{)n    ■    n){iii    '"  2n).r'    ■    ■■■ 


DIVEKGENT  SERIES  AND  CONTINUED  FRACTIONS.     ITo 

in  fractioneiu    contiuuam.     Nova  acta  Acad,  seieutiarum  im- 
perialis  Petropolitana?,  vol.  2,  pp.  86-45,  178-1. 
Gergonne.     (c)  Kecherches  sur  les  iVactions  continues.     Gergonue'a 
Annales  de  Math.,  vol.  9,  pp.  261-270,  1818. 

47.  Laplace.      (a)  Traite  de  mecanique  celeste.     Oeuvres,  vol.   4,  pp. 

254-257,  1805. 

Jacobi.     ip)    De    fractione    continua    in    quam    integrale  I     e    dx 

evolvere  licet.  Jour,  fiir  Math.,  vol.  12,  pp.  346-347,  1834,  or 
AVerke,  vol.  6,  p.  76. 

See  also  p.  79  of  No.  20,  and  the  first  note  under  lecture  2. 

O.  Periodic  Continued  Fraction-'^,  and  Continued  Fractions  Connected  with 
the  Theory  of  Elliptic  l<unctionH. 

48.  Abel,     (rt)  Sur  1" integration  de  la  formule  difFerentielle /jdx-/i/-K,  R 

et  p  etant  des  fonctions  entieres.     Jour,  fiir  Math.,  vol.  1,  pp. 
185-221,  1826,  or  Oeuvres,  vol    1,  p.  104  if. 
Dobma.     (/>)  Sur  le  developpemenc  de   v'^  en   fraction  continue. 
Nouvelles  Ann.  de  Math.,  ser.  3,  vol.  10,  pp.  134-140,  1891. 

49.  Jacobi.     (a)  Note  sur  une  nouvelle  application    de   I'analyse    des 

fonctions  elliptiques  a  I'algebre.     Jour,   fiir  Math.,  vol.   7,  pp. 
41-43,  1831,  or  Werke,  vol.  1,  p.  327. 
Borchardt.     [h)    Application    des   transcendautes   abeliennes   a    la 
theorie  des  fractions  continues.     Ibid.,  vol.  48,  pp.  69-104,  1854. 

50.  Tchebychef.     Sur  1' integration  des  difierentielles  qui    contiennent 

une  racine  carree  d'un  polynome  du  troisieme  ou  du  quatrieme 
degre.  Memoires  de  I'Acad.  imp.  des  sciences  de  St.  Peters- 
bourg,  ser.  6,  vol.  8,  pp.  203-232,  1857. 

51.  Frobenius  und  Stickelberger.     Teber  die  Addition  und  Multiplication 

der  elliptischen  Functionen.  Jour,  fiir  ]Math.,  vol.  88,  pp.  146- 
184,   1880. 

52.  Halphen.     Sur  les  integrales  pseudo-elliptiques.     Comp.  Rend.,  vol. 

106  (1888),  pp.  1263-1270. 

53.  Bortolotti.     Sulle  frazioni  continue  algebriche  periodiche.     Rendi- 

conti  del  Circolo  Mat.  di  Palermo,  vol.  9.  pp.  136-149,  1895. 
See  also  Nos.  21,  26(a),  40. 


H.   Miscellaneous. 
54.  Euler.     («)  Speculationes  super  formula  integrali 

x"d.r 
i  a'-  —  '2bx  —  ex'- 


/, 


17G  THE   BOSTON   COLLOQUIUM. 

iibi  simul  egregiiie  observationes  circa  fractiones  continuas  occur- 
rent.  Acta  Acad,  scientiariuii  imperialis  Petropolitanae,  1784, 
pars  posterior,  pp.  62-84,  1782. 
(b)  Summatio  fractionis  contiuufe  cujus  indices  progressionem 
arithmeticam  constituunt.  Opuscula  Analytica,  vol.  2,  pp.  217- 
239,  1785. 
55.  Spitzer.     (a)  Darstellung  des  iiiiendlicheu  Kettenbruchs 

^  ''"  X  '+1  H-  .iTf  "2  +  x  +  3  +  "  ' 

in  geschlossener  Form,  nebst  anderen  Bemerkungen.     Archiv 
der  Math,  und  Phys.,  vol.  30,  pp.  81-82,  1858. 
(b)  Darstellung  des  unendlichen  Kettenbruchs 

1  1  1 

Ox  +  1  H 


2.r  +  3  +  2x  +  5  +  2x  +  7  + 

in  geschlossener  Form.     Ibid.,  vol.  30,  pp.  331-334,  1858. 
(c)  Note  iiber  eine  Kettenl)riiche.     Ibid.,  vol.   33,   pp.   418-420, 

1859. 
(J)  Darstellung  des  unendlichen  Kettenbruches 

^'  ^         ^       '     ^   '    n{2x  +  3)  +  n{2x  +  5)  + 

in  geschlossener  P"'orm.     Ibid.,  vol.  33,  pp.  474-475,  1859. 

56.  Laurent,     (a)  Note  sur  les  fractions  continues.     Nouvelles  Ann.  de 

Math.,  ser.  2,  vol.  5,  pp.  540-552,  1866. 
This  treats  the  continued  fraction 

XXX 

1  +  T   i    T   r  '"' 
E.    Meyer.       (b)     I'ebcr     eine     Eigenscliaft     des      Kettenbruches 

.)•  —  ••■.     Archiv  der  ^Nlatli.  und  Plivs.,  ser.  3,   vol.  5, 

.)•  —  X  — 

]..  2S7,  liXK). 

Meyer's  results  will  be  found  on   p.  548  of  Laurent's  memoir 

and  (lilfcrs  only  in  that  x  has  been  replaced  by  —  \/x-. 

57.  Schlomilch.      (a)    Ucbcr  den    ]\ettenbruch  fiir  tan  z.     Zeitsclirift  fiir 

Math.  un<l  I'liys.,  vol.   16,  jip.  25!)~2(i(),   1S71. 
Glaisher.      {//)    \    continued    fraction    for    tan    nx.       M(!ssenger    of 

.Malh.,  ser.  2,  vol.  3,  p.    1.".7.    1S74. 
(c)    Xolc  oil   coiitiuucii  fi'iictions  for   tan    it.r.      liiid..  ser.  2,  vol.  4, 

jip.  (;.")~r),s.  1S75. 


DIVERGENT  SEIUES  AND  CONTINUED  FKACTTONS.     177 

58.  Schlomilch.     I'eber  die  Kettcn])nu'ln-'ntwickeluiig  fi'ir  unvollstiiii- 

dige  Ciamma-fuiu'tion.     Zeitsclnift  fi'ir  INIath.   und   Pbys.,   vol. 
16,  pp.  261-2G2,   1871. 

This  gives  tlie  development  of  I      f^"^  e~'df. 

59.  Schendel.     Ue])ereine  Kettenl)niehent\vickelinig.    Jour.  fiirMath., 

vol.  80,  pp.  95-96,  1875. 

60.  Lerch.     Note  sur  les  expressions  (pii.  dans  divcrscs  parties  du  plan, 

representent  des  fonctions  distinetes.      Bull,  des  sciences  Math, 
ser.  2,  vol.  10,  pp.  45-49,  1886. 

61.  Stieltjes.    (a)  Sur  qiiel(}ues  integrales  definies  et  leur  developpement 

en  fractions  continues.    Quar.  Jour,  of  pure  and  applied  Math., 
vol.  24,  pp.  870-382,  1890. 
(i)  Note  sur  cjuehiues  fractions  continues.     Ibid.,  vol.  25,  pp.  198- 
200,  1891. 

62.  Hermite.     Sur  les  polynomes  de  Legendre.     Jour,  fiir  Math.,  vol. 

107,  pp.  80-83,   1891. 

This  connects  D''^'P'''^\x)  -with  a  continued  fraction. 

IV.  On  the  Connection  of  Continued  Fractions  with  Differen- 
tial Equations  and  Integrals. 

A.   ii(cro//'.s-   Difffrcntkil  Equation. 

63.  Euler.     (o)  De  fractionibus  continuis  observationes.     Commentarii 

acadenii.'e    scientiarum    imperialis    Petropolitante,  vol.   11.  see 
pp.  79-81,  1739. 
{}))  Analysis  facilis  »([uationem  Kiccatianam  per  fractionem  con- 
tinuam  resolvcndi.     Memoiies  de  1'  Acad,  imperiale  des  sciences 
de  St.  Tetersbourg,  vol.  6,  pp.  12-29,  1813. 

64.  Lagrange.     Sur  I'usage  des  fractions  continues  dans  le  calcul  inte- 

gral.    Nouveaux  ]Mem.  de  I'Acad.  roy.  des  sciences  et  belles- 
lettres  de  Berlin,  1776,  pp.  236-264,  or  Oeuvres,  vol.  4,  p.  301  fi; 
One  of  the  iew  impoi'tant  early  works. 
See  TAh  ;  also  No.  i')Qa  for  work  on  differential  ecjuations  of  the  1st  order. 

B.    Mii?ci'Uancovf<  Differential  F.qvationf^  of  the  Second  Order. 

In  a  numerous  class  of  continued  fiactions  the  denominators 
of  the  convei'gents  satisfy  allied  (Heun,  ••  yleicligruppifje")  (\iffer- 
ential  equations  of  the  second  order.  Early  instances  are  found 
in  works  of  Gays!<  (No.  114),  Jacobi  (No.  65)  and  Heine  (No.  72). 
The  theory,  from  two  different  aspects,  is  furthest  developed  in 
66rt  and  76. 


178  THE    BOSTON    COLLOQITIUM. 

65.  Jacobi.      riitersucliunu'  i'lhor  die  DiU'erentialgleichung  der  hyper- 

froometi'isclien  Keihe.  Nachlas^;.  Join-.  fi'ir^Iath.,  vol.  56,  1859  ; 
see  in  particular  i;  8,  jij).  160-161.  or  Werke,  vol.  6.  p.  184. 

66.  Laguerre.     (a)  Sur  la  reduction  en  fractions  continues  d'une  frac- 

tion (jui  satisfait  a  une  e(|uation  ditlerentielle  lineaire  du  pre- 
mier ordredont  les  coellicients  sont  rationnels.  Jour,  de  Math., 
ser.  4,  vol.  1,  pp.  135-165,  1885. 

Tliis  is  a  comprehensive  memoir  which  incorporates  substan- 
tially all  the  following  memoirs  : 
(h)  Sur   la    reduction  en    fractions    continues    d'une    classe    assez 
etendue  de  fonctions.      C'omp.  Rend.,  vol.  87  (1878),  p.  923,  or 
Oeuvres.  vol.  1,  p.  322. 

(c)  Same  title  as  (a),  liull.  de  la  Soc.  Math,  de  France,  vol.  8 
(1880),  pp.  21-27.  or  Oeuvres.  vol.  ].  p.  43S. 

(d)  Sur  la  reduction  en  fraction  continue  d'une  fraction  qui  satis- 
fait a  une  equation  lineaire  du  pi-emier  ordre  a  coefficients  ration- 
nels. Comp.  Rend.,  vol.  98  (1884).  pp.  209-212  or  Oeuvres. 
vol.  1,  p.  445. 

67.  Laguerre.     (a)  Sur    rapi)roximation    des   fonctions  d'une    variable 

a\i  moyen  de  fractions  I'ationnelles.  Bull,  de  la  Soc.  Math,  de 
France,  vol.  5  (1877),  i)p.  78-92  or  Oeuvres,  vol.  ].  ]).  277. 

(b)  Sur  le  developpement  en  fraction  continue  de 

,"■ (7)^r,::.• 

Ibid.,  vol.  5  (1877).  pp.  i)5-99  or  Oeuvres,  vol.  1,  p.  291. 

(c)  Sur  la  ionction  I  I   . 

Il)id..  vol.  8  (1879),  ])]•.  :;(;-52,  or  Oeuvres,  vol.  1,  p.  345. 

(d)  Sur  la  reduction  en  fractions  continues  de  r'-",  F{.r)  dt'sig- 
nant  un  j)()lyn6nie  enlier.  .Tour,  de  Math.,  ser.  3.  vol.  6  (1880), 
]>]).  9'.»    llo.  or  Oeuvres,  vol.  1,  ]).  3)25. 

(d')  Same  subject.    ( 'oniji.  Renil. .  vol.  87  (1878).  ]).  820,  or  Oeuvres, 
vol.  1,  ]<.  :;]s. 
6S.   Humbert.      Sui'  la   i-rductiou   in  fractions  cont  inues  d'unt' classe  de 
foiietioii>.      I'.ull.  de   la   Soc.   .Math,   de    l-'rance,  vol.  8,  pp.   182- 
1S7,    1879    18S0. 
69.   Hermite  et    Fuchs.      Sur    un    <ieveloiiiiement    en    fraction    continue. 
Acta  Math.,  vol.    I.  p]L  89-92,  18S1, 
Si'f  also  No.  2u.  :',4  /'/■,  71-  7G. 


DIYElKiENT  SKKIKS  AND  CONTTNTKl)  FKACTIOXS.     179 

C.    Differential  IjjiKdions  of  Order  Higher  than  the  Seeond. 
70.   Pincherle.     Sur    la    ,uriu'nitioii    de    systt'mes   rci'urrcnts  au   movcn 
cruiu"  ('([nation  liiu'aire  diU'erentielle.      Acta  ]\ratli.,  vol.  KJ,  pp. 
341-80;^  1892-8. 
Sec  also  No.  15.  86.  87.  12ib. 


.^''fU)dx 
D.     The   integral  |    -'^  -'      • 


integral  I    ' 


71.  Heine,     (a)  reber  Krttenbrik'lie.     ]Nroiuitsl)eiiclit(>  der  k.  prenssi- 

scluMi  Akad.  der  Wisseusehaf'ten  zn  Berlin.  ISGO,   pp.  436-451. 
(«')  31ittheilnng  iiber  Kettenbriielie.     Auszug  aus  dem  Monatsbe- 
ricbte,  n.  s.  av.     Jour,  fur  ^lath.,  vol.  67,  pp.  315-326,  1867. 
See.  also  Nos.  12.  26a,  28,  45,  102,  113,  118o. 

E.   nyperellij)tic  and  Similar  Ahelian  Integrals. 

72.  Heine.     Die    Lame'schen    Functionen   verschiedener    Ordnungen. 

Jour,  fiir  Math.,  vol.  60,  1862,  pp.  252-303  ;  in  particular  pp. 
256,  275,  294-297.  Or  see  his  Handbuch,  vol.  1  (2'^  Auf.),  pp. 
388-396  and  468. 

73.  Laguerre.     Sur  I'approximation  d'une  classe  de  transcendantes  qui 

coinpreuncnt  comnie  cas  particulier  les  integrales    hyperellixi- 
ticiues.     CoBip.  Rend.,  vol.  84,  pp.  643-645,  1877. 
(Not  found  in  vol.  1.  of  his  Oeuvres.) 

74.  Humbert.     Sur  I'equation  dilk'rentielle  lin(''aire  du  second  ordre. 

Jour,  de  I'T^c.  Polytech.,  vol.  29,  cahier  48,  pp.  207-220,  1880. 

75.  Heun.      («)  Die  Kugelfunctionen  und  Lame'schen  Functionen  als 

Determinanten.     Dissertation,  pp.  1-32,  Gottingen,  1881, 
(l>)  I'eber  lineiire  Differentialgleichungen  zweiter  Ordnung  deren 

I.i(")sungen  (lurch  den  Kettenbruchalgorithmus  verkniipft  sind. 

Habilitationsschrift.      1881. 
(c)  Integration  regularer  linearer  DiHerentialgleichungen  zweiter 

Ordnung  durch  die  Kettenbruchentwickelung  von  gauzen  Abel'- 

schen  Integralen  dritter    Gattung.     Math.   Ann.,  vol.  30,  pp. 

553-560,  1887. 
{d)  P>eitr;ige   zur    Theorie    der    Lanu''schen    Functionen.      Math. 

Ann.,  vol.  33,  i)p.  180-196,  1889. 

The  important  group-properties  of  the  continued  fraction  are 

liere  brought  out  and  are  further  developed  in  No.  7(). 

76.  Van  Vleck.     Vaxy  Kettenbruchentwickelung  hyperelliptischer  und 

ahnlicher  Integrale.  Dissertation,  Gottingen;  published  in  the 
Anier.  Jour,  of  :Math.,  vol.  16  (1894),  pp.  1-91. 


180  THE    ]«)ST()X    COLLOQUIUM. 

After  development  first  from  an  alaebi'aie  .'•tai)di)oint  the  sub- 
ject is  carried  furtlier  by  tlie  metliod  of  conformal  representation. 
Tlie  su<i:gestion  of  this  treatment  is  given  iu  Klein's  Differen- 
tialgleiehungen,  1890-91,  voL  1,  pp.  1<S0-18(). 

V.      CiENERALTZATION    OF    TJIK    AlcEBRAIC    CONTINUE])    FRACTION. 

A.    General  Theory. 

So  far  as  I  have  been  able  to  ascei'tain,  the  first  instance  of  the 
generalization  is  contained  in  Hermife's  memoir,  Xo.  84.  The 
development  of  a  general  theory  is  due  to  Fade  and  Fincherle. 
Xos.  77a,  77/>.  and  79a  are  especially  recommended. 

77.  Pincherle.     (a)  Saggio  di  tina  generallizzazione  delle  frazioni  con- 

tinue algebriche.     IMemoirie  della  R.  Accad.  delle  Scienze  dell' 

Istituto  di  Bologna,  ser.  4,  vol.  10,  p.  513-538,  1890. 
(a')  Di    un'estensione    dell'    algorithm©    delle    frazioni    continue. 

Rendiconti,  R.   Istituto  Lombardo  di  Scienze  e  Lettere,  ser.  2, 

vol.  22,  pp.  555-558,  1889. 
(b)  Sulla  generalizzazione  delle  fi'azioni  continue  algebri([ue.     Au- 

nali  di  :Mat.,  ser.  2,  vol.  19.  pp.  75-95,  189 L 

78.  Hermite.     Siir  la  generalisation  des  fractions  continues  algebri(pies, 

Annali  di  :\Iat.,  ser.  2,  vol.  21,  pp.  289-308,  1893. 

79.  Pade.     (a)    Siu-    la    generalisation    des    fractions    continues    alge- 

briqucs.     .Jour,  de  Math.,  ser.  4,  vol.  10,  pp.  291-329.  1894. 
(a')  Same  subject.     Com]).  Rend.,  vol.  118,  p.  S4S,  1894. 

80.  Bortolotti.     I^n  contributo  alia  teoria  delle  ibrme  linear!  alle  ditfer- 

enze.      Annali  di  3Iat.,  ser.  2,  vol.  23.  p]).  309-344,  1895. 

81.  Cordone.      Sopra  uti  ])r<jblcma  fuiidamentalc  delle   teoi'ia  delle  fra- 

zioni continue  algebriche  geueralizzate.      Rendiconli  del  Circolo 
di  raUinu),  vol.  12.  p]).  240-257,  1898. 

Cordone  seeks  the  regular  algorithms  which  are  similar  to 
tliose  of  I'adr  but  occur  in  connection  with  //  series  in  descend- 
ing ]io\vcrs  of  .)•. 

/.'.    C'tHU'eri/f/ice  <if  the  Generalized  Ahjoriihtn. 

82.  Pincherle.      Contributo  alia  generalizzazione  delle  frazioni  continue. 

Mcmoii-ic  della  K.  Ac<'ad.  delle  Scienze  dell'  Istituto  di  I'ologna. 
^ei'.  5.  vol.  4,  ))]).  297-;!2(».   ISHl. 

83.  W.   Franz  Meyer.      (>()    rdx-r    kettentiruchahnlichen    Algorithmi'ii. 

\'ciband.  des  erstcn    intrrnationalen    Matheniatiker-Kongresses 
in  Ziiiich,  \>\k    KlS    ISL   ISDS  ;  see  in  jiaiticular  i;  7. 


DIVEKGEXr  SEKIKS  AND  COXTIXrEl)  FK ACTIONS.     181 

(aQ  Zur  Tlieoi'ie  dvv  kelleiibnu'lKiliiiliclKMi  Aluoritlnueii.  Sclirif- 
ten  (ler  pliys-iikonoinisclieii  Crosellschiifl  zu  KiHiiiislxTt;-,  vol. 
08,  i)p.  57-0(5,  1897. 

C.  Special  C<i!<es  of  thf  AUjorithm. 

84.  Hermite.     Hur  la  Ibnction  exi)oneiitielle.     Comp.  Rend.,  vol.  77,  pp. 

18-24,  74-79,  22()-233,  285-293,  187:5. 

This  is  the  famous  work  proving  tlie  transcendenee  of  e. 

85.  Hermite.     (a)  Sur   I'expression    f/ sin  .r  —  T  eos  x  ■,-■  IT.      Extrait 

(Tune  lettre  a  ]Monsieur  Paul  Gordan.  .Four,  fur  ^latli.,  vol. 
76,  pp.  303-311,  1873. 

(b)  Sur  quehiues  approximations  algel)ri((ues.  Iltid.,  vol.  76,  pp. 
342-344,  1873. 

(c)  Sur  quel([ues  ('([uations  ditlerentielles  lineaires.  l-lxtrait  d'une 
lettre  a  M.  L.  Euchs  de  Gotlingue.  Ibid.,  vol.  79.  pp.  324-338, 
1875. 

86.  Laguerre.     Sur  la  foactiou  expouentielle.     Bull,  de  la  Soe.  ^lath.  de 

France,  vol.  8  (1880),  pp.  11-18,  or  Oeuvres,  vol.  1.  p.  336. 

87.  Humbert,     (a)  Sur  une  generalisation  de   la  theorie  des  fractions 

continues  algebri(iues.     Bull,  de  la  Soc.  IMath.  de  France,  vol. 
8.  pp.  191-196:  vol.  9.  pp.  24-30.  1879-1881. 
(b)  Sur  la  fonction  (.»•  —  1)'.     Ibid.,  vol.  9.  pp.  56-58.  1880-81. 

88.  Pincherle.     Sulla  rappresenta/ioue  approssimata   di  una  fuuzione 

mediantc  irrazionali  quadratic!.  Rendiconti,  R.  Istituto  Lom- 
bardo  di  Scienze  e  I^ettere,   ser.  2,  vol.  23,  i)p.  373-376,  1890. 

89.  Pincherle.      (a)    Una    nuova    esteusioue    delle     funzioni    sferiche. 

^lemoii'ie  della  R.  Accad.  delle  Scienze  delTI^tituto  di  Bologna, 
ser.  5.  vol.  1,  pp.  337-370,  1890. 

(«')  Sulla  generalizzazione  delle  funzioni  sferiche.  Bologna  Ren- 
diconti, 1891-92,  pp.  31-34. 

(&)  Vn  sisteina  (rintegrali  ellittici  considerati  come  funzioni 
deirinvariante  assoluto.  Atti  della  R.  Accad.  dei  Liucei,  ser. 
4.  vol.  7,,  PI).  74-80,  1891. 

90.  Bortolotti.     (<()  Sui  sistemi  ricorrenti  del  3°  ordine  ed  in  particolare 

sui  sistemi  })eriodici.      Rendiconti  del  C'ircolo  di  Ralermo,  vol.  5, 
pp.  129-151,  1891. 
(b)  Sulla  generalizzazioiu^  delle  frazioni  continue  algcbriche  peri- 
odiche.      Ibid.,  vt>l.  (k  pp.  1-13.  1892. 

]'I.    Series  of  ro/y/wmidl.s  {Xdlierir/u/stiniuer). 
The  series 


182  THE   BOSTON   COLLOQrir:\[. 

was  first  given  by  Ilciac  in  Crelle's  Jour.,  vol.  42  (1851),  p.  72. 
See  also  his  Handljuch,  vol.  1,  p}).  78-79,  197-200.  Among 
the  numerous  works  relating  to  expansions  in  terms  of  Kugel- 
functionen  rrster  und  zweiter  Gattung  may  be  mentioned  : 

91.  Bauer.     A'on  den  Coeilicienteu  der   Reihen  von  Kugelfunctionen 

einer  Variablen.     Jour,  fur  Math.,  vol.  56,  pp.  101-121,  1859. 

92.  C.  G.  Neumann.     Ueber  die  Entwickelung  einer  Function  mit  imag- 

injirem  Argumente  nach  den  Kugelfunctionen  erstcr  und  zweiter 
Gattung,  Halle.  1802. 

93.  Thome.     Ueber    die  Keiheu    welche    nach    Kugelfunctionen    fort- 

schreiteu.     Jour,  fiir  Math.,  vol.  06,  pp.  837-3-18,  1866. 

94.  Laurent.     Memoire  sur  les  fonctions  de  Legendre.     Jour,  de  ^Nlath., 

ser.  8,  vol.  1,  pp.  378-398,  1875. 

See  the  comments  by  Heine  in  vol.  2,  pp.  155-157,  also  by 
Darboux  and  Laurent  in  the  same  vol.,  pj).  240,  420. 

Numerous  memoirs  relate  to  series  in  terms  of  the  polynom- 
ials arising  from  the  expansion  of  (1  —  2a.c  —  «'')'•  It  suffices 
here  to  refer  to  the  Encyklopiidie  der  ]Math.  Wissenschaften, 
I  A  10,  §31. 

95.  Frobenius.     Ueber  die    Entwicklung    analytischer    Functionen    in 

Keihen,   die  nach    gegebenen  Functionen  fortsclireiten.     Jour, 
fiir  3rath.,  vol.  73,  pp.  1-30,  1871. 
An  interesting  memoir. 

96.  Darboux.     Sur  Tapproximation  des  fonctions  de  trcs-grands  nom- 

bres  et  sur  uiie  classe  etendue  de  develo})pements  en  serie.  Part 
2.     Jour,  de  Math.,  ser.  :'.,  vol.  4,  i)p.  377-416,  1S78. 

97.  Gegenbauer      I'el)er  Kettenbriiche.    Wiener  Bericlite,  vol.  SO,  Abth. 

2.  j.p.  7<i3-775,  18S0. 

98.  Poincare.      (n)  Sur  les  (Mpiations  lincaires  aux  dilicreiitielles  ordi- 

naircs  et  aux  dillrreiu'es  tinies.  Ainer.  .lour,  of  ]Math.,  vol.  7, 
J. p.  248   257,  1SS5. 

'I'iiis  gives  an  important  criterion  for  tlic  convergence  of  series 
of  polynomials.      See  h'cture  4. 
((//)   Sur  les  series  des  polynomes.      ('omj).   Kend..  V(j1.  56,  p.  687, 
iss;;. 

99.  On  the  series  i.l,/.'-       '',)(.'•    -  n.)  ■  ■  ■  (r  —  t(  ). 

A  series  of  this  loi-iu  is  emph)yed  in  Newton's  interixiiation 
fornuda,  l'hiioso])lii;e  naturalis  piiiieipia.  l>oolc  3,  leninia  V. 
Seethe  Ihiey  1< lopiiij ie  der  .Math.  \Vi<senschaften.  I  I)  8,  i;  8.  A 
.-similar  u-e  is  made  h\' 


DIYEKGENT  SKKIKS  AND  CONTINIEI)  FRACTIONS.     183 

Cauchy.     (a)  Sur  les  fonctions  iiiterpoluires.     Comp.    Rend.,   vol. 
11,  pp.  775-789,  18-11. 
See  next  Xo.  95. 

Peano.     (6)  SuUe  funzioni  interpolari.     Atti  della  R.  Acctul.  delle 

Scieiize  di  Torino,  vol.  IS,  pp.  573-580,  1S8.'5. 
Bendixson.     (c)  Sur  une  exteuslou  a  riiifini  de  la  forraule  d"inter- 

l)o]ation  de  Gauss.     Acta  Math.,  vol.  9,  pp.  1-34,  1886. 
(c')  Sur  la   formule  d' interpolation  de  Lagrange.     Gonip.  Rend., 

vol.  101  (1885),  pp.  1050-1053  and  1129-1131. 
Pincherle.    (d)  SuU'interpolazioiie.     Memoirie  della  R.  Acead.  delle 
Scienze  di  Bologna,  ser.  5,  vol.  3,  pp.  293-318. 

(See  a   "note  liistorique " '    by  Enestrom,  Comp.    Rend.,  vol. 
103,  p.  523,  1886). 
See  also  No.  103. 

100.  Pincherle,     Sur   le  developpement  d'une  fouction   analytique    en 

serie  de  polynomes.     Comj).  Rend.,  vol.  107,  p.  986,  1888. 

101.  Pincherle.     Resume  de  quelques  resultats  relatifs  a  la  theorie  des 

systemes  recurrents  de  fonctions.     Mathematical  Papers,  Chi- 
cago Congress,  1893,  pp.  278-287. 

102.  Blumenthal.     Ueber  die  Entwickelung  einer  willkiirlichen  Funk- 

tiou  nach  den  Nennern  des  Kettenbruches  fur 


£ 


c^(c)c7^ 


Dissertation,  Gottingen,  1898. 

The  most  advanced  development  of  this  subject  is  found  in 
the  work  of  Bltunenthal  and  Pincherle. 

103.  Laurent.     Sur  les  series  de  polynomes.     Jour,    de  Math.,  ser.    5, 

vol.  8,  pp.  309-328,  1902. 

104.  Stekloff.     Sur    le  d('veloj)pement  (ruiu'  f^juction  donee  en    series 

procedant  suivant  les  polynomes  de  Tcht'biclietl' et,  en  ])articul- 
ier,  suivant  les  polynomes  de  Jacol)i.  Jour,  fi'ir  ]Math..  vol. 
125,  i)p.  207-236,  1903. 

See  also  Xos.  20,  70.  71. 

104  bis.  Rouche.  3I('moire  sur  le  drveloppement  des  fonctions  en  series 
ordonnces  suivant  les  denominaleurs  des  rcduites  d'uiu'  frac- 
tion continue.      Jour,  de  I'Ec  Polytt-ch. ,  cahier  37,  pp.  1-34. 

Tliis  memoir  lias  a  chjse  connection  with  the  work  of  Tclieby- 
chcf. 

VII.    On  the  floot.-i  of  the  Xinnerators  and  Denominators  of  the  Conrergents. 

105.  Sylvester,     (a)  On  a  remarl<able  modification  of  Sturm"s  theorem. 

I'liil.  3Iag.,  ser.  4,  vol.  5.  ])p.  446-456.  1853. 


184  THE    BOSTON    COLLOC^rirM. 

(/>)  Note  on  a  remarkable  mollification  of  Stnrm's  theorem  and  on 

a  new  rule  for  finding  snj)eri()r  and  inferior  limits  to  the  roots  of 

an  e(|uation.      Ibid.,  vol.  6.  pp.  14-20,  ISoo. 
(c)  On  a  new  rule  for  finding  superior  and  inferior  limits  to  the 

real  roots  of  any  algebraic  eciuation.      Ibid.,  vol.  G,  ])p.  138-140, 

1853. 
((/)  Note  on  the  new  rule  of  limits.     Ibid.,  vol.   (>,  pp.    210-213, 

1853. 
(i?)  On  a  theory  of  tlie  syzygetic  relations  of  two  rational  integral 

functions,  comprising  an  ai)plication  to  the  theory  of  Sturm's 

functions,  and  that  of  tlie  greatest  algebraic  ct)mmon  measure. 

Phil.  Trans.,  1853  ;  see  in  i)articular  ]>.  4!(G  11". 
(/)  Theoreme  sur  les  limitesdes  racines  reelles  des  ecpiations  alge- 

bri(iues.     Nouvelles  Ann.  de  Math.,  ser.  1,  vol.  12,  pp.  286-287, 

1853. 
(//)  Pour  trouver  une  limite  superieure  et  une  limite  inferieure  des 

racines  reelles  d'une  equation  (luelcoiKiue.     Ibid.,  ser.  1,  vol.  12, 

pp.  329-330,  1853. 
106.  Laguerre.     Sur  ([uehpies  })roprietes  des  ecpiations  algebricpies  (pii 

out  toutes  les  racines  reelles.     Nouvelles  Ann.  de  Math.,  ser.  2, 

vol.  1!)  (1880),  pp.  224-239,  or  Oeuvres,  vol.  1,  pp.  113-118. 
Laguerre  considers  here  the  roots  of  the  numerators  and  de- 

nominatoi-s  of  the  ap]iroximants  for/(j-)  and  l^f{x)^\■hvu  f{x)  is  a 

polynomial  with  real  roots. 

107.  Gegenbauer.     (a)   Tebcr  algebraische  (rleichungen  welche  nur  reele 

Wuiv.eln  l)esitzen.      Wiener  Berichte,   vol.    84   (1882).    Abt.   2, 
see  in  particular  j)]).  1106-1107. 
(/>)  Ueber  algebraische  Cileicluingen  welclie  eiue  i)estimmte  An- 
zahl  comj)lexer  Wulzeln  besitzen.      Ibid,  vol.  87.  pp.  264-270, 
18S3. 

108.  Markoff.     Sur    les   racines    {]v    certaines   ('(luatit)ns.      ]Math.    Ann., 

vol.  27,   pp.  143-150,  18S6. 
10.S/>/s.   Hurwitz.      I'cIxm-    die    Xullstellen    der    Bessel'scheii    I'luiction. 

."\Iatli.  Ann.,  vol.  33,,  j.]).  2l6-2t;6,   18S9. 

Although   tlic  functions  considered   in   this   memoir  are  of  a 

special    charactc]',  tlu'  memoir  is  mentioned    here  on  ac<.'ount   of 

t  he  metluxls  employed. 
1(19.    Porter.      ( )n  the  roots  ol' fuiict  ions  connected  by  a    linear  recurrent 

relntidii  of  the  secdiid  order.      Annals   ol"   Matli.,  ser.  2.  \dl.  3, 

]>\>.  5."')   71  >,   i'.iO'J, 
Srr  (i/n,,  S<>>.  2n.  2tWr  .".I,  :'.•_^^  45.  5<;,  71.  74,  76.  S7w.   1  1  S,( 


DIVEUGKXT  SKRIKS  AX])  CONTIXrKD  FKACTIOXS.     18J 

J'lII.     Apjjroximafio/i  to  a  Function  at  yiarc  Thmi  0/>e  I'oint.      Coniwction 
of  Conlinved  Fractions  )ritli  the  TJieory  of  Interpolation. 
I'lider  Xo.  1)9    have    ))een  already  classilied    various  works 
wliieh  I'clatc  to  simultaneous  a])proxiination  at  several   points. 
In  addition,  the  followino:  memoirs  may  also  he  consulted: 

110.  Cauchy.     Sur    la    formule    de    Lagrani^e    relativ    a    interpolation. 

Analyse  Alg.,  p.  028,  or  Oeuvres,  ser.  2.  vol.  8,  pp.  429-433. 

111.  Jacobi.     Ueher  die  Darstellung  eine  Keihe  gegehner  Werthe  (lurch 

eine  gehrochene  rationale  Function.     Joui'.  fiir  3Iath..  vol.  30, 
pp.  127~lo6,  1846,  or  Werke,  vol.  3,  p.  479. 

112.  Pade.     Sur  Textension  dcs  proprietes  des  reduites  d'une  fonction 

aux   fractions  d'interpolation   de  Cauchy.     Comp.  Keud.,   vol. 
130.  p.  697,  li)00. 
See  also  Xos.  U'-),  99. 

For  general  works  upon  interpolation  which  hring  out  the 
relation  of  the  subject  to  continued  fractions,  see  Heine's 
Handhuch  der  Kugelfunctionen,  vol.  2,  and  ^Markolfs  Difiier- 
enzenrechnung  (deutsche  I'ebersetzung),  chap.  1,  6,  7  ;  also  the 
following  memoir  : 

113.  Posse.     Sur  (|uel([ues    applications  <les   fractions    continues    alge- 

bri(iues.      I'p.  1-175.  1886. 

114.  Gauss.     ^lethodus  nova  intt-gralium  valores  per  approximationem 

inveniendi.      Werke,  vol.  3.  pj).  165-196.  1816. 

115.  Christoffel.      I'eber  die  (raussische  Quadratur  und  eine  Verallge- 

meinerung  dei'selhen.     .Tour,  fi'ir  ^Nlath..  vol.  55.  jip.  61-82.  1858. 

116.  Mehler.     P>emerkungen  zur  Theorie  der  median ischen  (^uadraturen. 

Ibid.,  vol.  63,  pp.  152-157,  1864. 

117.  Posse.     Sur  les  (quadratures.     Xouvelles  Ann.   de  IMath.,   ser.    2. 

vol.  14.  pp.  49-62,  1875. 
lis.  Stieltjes.     (a)  Quehpies  recherches  sur  la  tln'orie  des  (piadratures 
dites  m(''cani(iues.      Ann.  de  I'l-x'.  X'or.,  ser.  3,  vol.  1.  jip.  409- 
426,  1884. 

We  find  hei-e  the  origin  of  his  nota])le  1894  memoir,  X'o.  26(/. 
(a')  Sur    revaluation  a])proch('e    des    int('grales.     Comp.    IJend., 
vol.  97.  pp.  740  und  798,  1883. 


(b)  X'ote  sur  1"  iiit('grale   |   f{.r)G(.r)<Ix. 


Xouv.  Ann.  de  :\Iath.,  ser.  3,  vol.  7,  pp.  161-171,  1888. 
119.  Markoff.     Sur  la  m('-thode  dc  (fauss  pour  le  calcul  approclu''  des  in- 
t(''grales.      31ath.  Ann.,  vol.  25,  pp.  427-432,  1885. 


I'^^'i  THE    BOSTON   COLLOQUIUM. 

120.  Pincherle.     Su  alciine  fonne  ;i])])ros8imato  i)er  la  rapprcsciitazionc 

(li  ruiizioni.     ^Icraoiric  della  Iv.  Accad.  delle  Seienze  delTIstitiito 
di  Bologna,  ser.  4,  vol.  10.  pp.  77-88,  1889. 

121.  Tchebychef.      A  hrief  sketch  ol'tlie  memoirs  below  will  be  found  on 

pp.  17-20  of  VassHicf's  memoir  on  "  P,  L.  Tchebychef  et  ."^on 
oeuvre  scientifique."' 

(a)  Siir  les  fractions  contimies.  Jour,  de  jNfath.,  ser.  2,  vol.  8,  pp. 
28i»-328,  1858,  or  Oeuvres,  vol.  1,  p.  208-230. 

(h)  Sur  line  formiile  d'analyse.  Bull.  Phys.  Math,  de  I'Acad.  des 
sciences  de  St.  P«'tersbourg,  vol.  18,  pi>.  210-211,  1854,  or  Oeuv- 
res. vol.  1,  pp.  701-702. 

(c)  Sur  une  nouvelle  serie.  Ibid.,  vol.  17,  pp.  257-2G1,  1858,  or 
Oeuvres,  vol.  1,  pp.  881-884. 

(d)  Sur  r interpolation  i)ar  la  methode  des  moindres  carres.  Mem. 
de  I'Acad.  des  sciences  de  St.  Peter.'^bourg,  ser.  7,  vol.  1,  ])p. 
1-24,  1859,  or  Oeuvres,  vol.  1,  pp.  473-498. 

(e)  Sur  le  developpenient  des  fonctions  a  une  seule  variable.  Bull, 
de  I'Acad.  imp.  des  sciences  de  St.  Petersbourg,  ser.  7,  vol.  1, 
I)p.  J  94-199,  1860,  or  Oeuvres.  vol.  1,  pj).  501-508. 

IX.    MlSCKLLANEOU.'^. 

122.  Tchebychef.     (a)  Sur  les  fractions  continues  algebriques.     Jour,  de 

:\Iath.,  ser.  2,  vol.  10.  pp.  853-358.  1865,  or  Oeuvres,  vol.  1,  pp. 
611-614. 

(b)  Sur  le  deve]oi)pement  des  fonctions  en  series  a  I'aide  des  irac- 
tions  continues,  18()().     Oeuvres,  vol.  1,  j)p.  617-636. 

{(■)  Sur   les  expressions  a])]irochees.   lineares  ])ar  I'ajtport   a  deux 

jiolynonies.      lUill.  des  sciences  ]Math.  et  Astron.,  sei'.  2.  vol.  1, 

]>\)^  2S9,  882  ;   1877. 
Hermite.      {<l)  Sui'  une  extension  doiiiiee  a  la   thcoi'ie  des  fractions 

continues   ])ai'  M.    Tcheliychef.      Jour,    fiir   INIath..   vol.   88,  pp. 

12-18.  1880. 

123.  Tchebychef.      (a)     Sur  les  valcui's  limites  des  integrales.      Jour,  de 

Matli..  ser.  2.  vol.  19.  ].p.   157-1<;0,  1874. 
(ft)  Sui-  la  re]iit'sentation  des  \aleurs  limites  des  integrales  pai' des 

roidus  integraux  (18S5).      Acta.  -Alatli.  vol.  9.  ])ji.  ;>5-56,  1887. 
Markoff.      (r)   1  )rin(instratioii  dc  ccitaincs  inrgalitt's  de  ]\I.  Tclieby- 

clirf.      Math.  Ann.,  vol.  21,  j.p.  172    178,   IS84. 
((I)    NdUNcllcs  ajijilicat ions  (les  fi'actions  continues.      ]\Iath.  Ann., 

vol.    17,  pii.  57!»-597,  1S!)(;. 

124.  Laguerre.      {it)   Siii'   !<•   d('\ cldiijienient    de  (j- ~   c)'"  sui\ant    les  juiis- 

>aiiee-    de    ( :. •-        1).      ('(imp.     Jiend.,     \'ol.    S6    (1878).   j).    956,  or 
<  hiix  i'e~.   \  111.    1 ,   p.   I'.l  5. 


DIVERGENT  SERIES  AND  CONTINFED  FRACTIONS.     187 

(b)  Sur  le  dt'vcloppement  d'une  fonction  suivant  les  puissances 
d'une  polynorae.  Jour,  fi'ir  Math.,  vol.  88  (1880) ;  in  particular, 
p.  37,  or  Oeuvres,  vol.  1,  p.  298. 

(c)  Same  subject.  Comp.  Rend.,  vol.  86,  (1878)  p.  383,  or  Oeuv- 
res, vol.  1,  p.  295. 

(d)  Sur  quelques  theoremes  de  M.  Hermite.  Extrait  d'une  lettre 
addressee  a  M.  Borchardt.  Jour,  fiir  Math.,  vol.  89  (1880),  pp. 
340-342,  or  Oeuvres,  vol.  1,  p.  360. 

125.  Sylvester.     Preuve  que  tt  ne  peut  pas  etre  racine  d'une  equation 

algebrique  a  coefficients  entiers.     Comp.  Rend.,   vol.   Ill,   pp. 
866-871,  1890. 

A  fundamental  error  in  the   proof  has  been  pointed  out  by 
IMarkotr.     See  p.  386  of  vol.  30  of  the  Fortschritte  der  Math. 

126.  Gegenbauer.     I'eber  die  Niiheruugsnenner  reguliirer  Kettenbriiche. 

Monatshefte  fiir  Math,  und  Phys.,  vol.  6,  pp.  209-219,  1895. 

127.  Bortolotti.     Sulla  rap])resentazione  approssimata  di  funzioni  alge- 

briche  per  mezzo  di  funzioni  razionale.     Atti  della  R.  Accad. 
(lei  Lincei,  ser.  5.  vol.  1,,  pp.  57-64,  1899. 

Addendum  to  I  A. 

128.  Euler.    De  fractionibus  continuis  dissertatio.   Comment.    Petrop., 

vol.  9,  ]).  129  fl'.,  1737. 


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